1. Introduction

Electromagnetically induced transparency (EIT) is a quantum interference effect in atomic system, which can dramatically change optical properties of medium [1]. Although the novel effect has tremendous potential applications,the practical application of EIT is strictly limited by the harsh conditions of preserving quantum state coherence [2] because of the sharp dispersion and violent resonance. Recently, plasmon-induced transparency (PIT), an EIT-like optical effect, based on metamaterial structures including cut wires [3–5], split-ring resonators [6–8], and coupled waveguide resonators [9] has drawn enormous attentions due to its outstanding and wide practical applications, such as sensors [10,11], optical storage [12], polarization conversion [13], and switches [14–16]. In general, there are usually two kinds of schemes to achieve the PIT effect: the bright–bright mode coupling [2, 17–21] and the bright-dark mode coupling [3, 8, 10, 22–25]. The former is based on the frequency detuning and weak hybridization of two bright modes, while the latter is based on the destructive interference between the bright mode and dark mode. For example, Zhang et al. [3] first proposed a specific design for the realization of the EIT-like system based on the near-field coupling between bright and dark modes. Kekatpure et al. [20] proposed the analogue of plasmonic EIT at optical frequency in a system of nanoscale plasmonic resonator antennas based on the phase coupling between bright modes.

However, most of these structures are composed of metallic materials, and the performance of surface plasmon in these metals is constrained because of the difficulties in controlling permittivity functions and the existence of enormous material losses, which will result in a low modulation range. As an alternative method, graphene has drawn great attention among researchers because of its dramatic optical properties including extreme field confinement, low propagation losses, especially the gate-voltage-dependent feature that the Fermi energy ($E_F$) of graphene can be tuned dynamically by using the external electrostatic gating. More recently, a novel state of quantum matter that can be considered as “three–dimensional (3D) graphene”—3D Dirac semimetals that are also called bulk Dirac semimetals, arousing great interests among researchers. Similar to graphene, the permittivity functions of 3D graphene can also be dynamically controlled by adjusting $E_F$. Comparing to graphene, a kind of one-atomic-layer material, 3D Dirac semimetal films (DSFs) are easier to process and more stable. Furthermore, because of the crystalline symmetry protection against gap formation [26–28] in DSFs, the mobility is ultrahigh, reaching 9 × 10⁶ cm² V⁻¹ s⁻¹ at 5 K [29], which is much higher than that in the best graphene (2 × 10⁵ cm² V⁻¹ s⁻¹ at 5 K) [30]. These features indicate that the DSFs can be considered as a new class of active plasmonic material and enable plasmonic devices that can be effectively tuned at different frequencies.

In this letter, we propose two parallel-coupled coplanar DSF strips based on the weak hybridization between two bright modes to obtain the PIT effect in the THz region. Both the upper and lower strips could be excited by the incident light individually, and a transparency window can be observed when they are placed together. The resonant frequency and line width can be adjusted actively by varying the Fermi energy of the DSF strips and changing the geometry parameters of the design. Finally, a group delay of more than 1.86 ps is achieved under different Fermi energies in our design. Undoubtedly, such a simple DSFs-based PIT system can be used for tunable THz functional devices, such as THz sensors, modulators, and slow-light devices.

2. Structure and method

A unit cell of the plasmonic metamaterial consisting of two parallel DSF strips with dielectric substrate is illustrated in Fig. 2. The refractive index (RI) of the substrate is considered to be 1.5 and the thickness of DSFs is set as 0.2$\mu m$. The size of the unit cell can be described by horizontal and vertical periodicities of $P_x=115\text{ μm }$ and $P_y=85 \text{μm}$, respectively. The incident waves are irradiated along the z direction with $E_x$ polarization. All the 3D Dirac semimetal plasmonic films are numerically investigated by using the time-domain solver of the CST microwave package. The complex conductivity of the Dirac semimetal is adopted from a random-phase approximation theory (RPA) [31]. Accounting to the intraband and interband contributed to the longitudinal, the dynamic conductivity of the Dirac semimetal can be written as:

where $\text{G}\left(\text{E}\right)=\text{n}\left(-\text{E}\right)-\text{n}(\text{E})$ with $\text{n}(\text{E})$ being the Fermi distribution function, $E_F$ is the Fermi level, $k_F=E_F/\hslash {\nu }_F$ is the Fermi momentum, ${\nu }_F={10}^6\text{m}/\text{s}$ is the Fermi velocity, $\text{ε}=\text{E}/E_F$, $\text{Ω}=\hslash \omega /E_F$, ${\epsilon }_c=E_c/E_F$, $E_c=3$ is the cutoff energy, and $g$ is the degeneracy factor. Here, we give the dynamic conductivity of the DSFs, as shown in Fig. 1. The dotted line stands for the real part of the dynamic conductivity, while the solid line stands for the imaginary part. As we can see from Fig. 1 that the real parts of the dynamic conductivities are 0 when $\hslash \omega /E_F$ is less than 2 and step signals are generated at $\hslash \omega /E_F=2$. As for the imaginary parts, the three curves intersect at one point at $\hslash \omega /E_F=1.23$.

 

Fig. 1 Schematic illustration of two parallel DSF strips on a substrate with RI 1.5. The geometric parameters are ${\text{L}}_2=32 \mu \text{m}, \text{w}=5 \mu \text{m},\text{d}=10 \mu \text{m}.$ Here, the literal displacement s and length of the upper strip ${\text{ L}}_1$ are variable.

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Correspondingly, using the two-band model and taking into account the interband electronic transitions, the permittivity of the 3D Dirac semimetals can be expressed as:

Where ${\epsilon }_b=1$ for $\text{g}=40$ (AlCuFe quasicrystals [32]) and ${\sigma }_0$ is the permittivity of vacuum. According to the above permittivity equation of Dirac semimetals, values of permittivity under different frequencies could be calculated by Matlab, and import these dispersion values to the characteristics of the new material to accomplish the modeling.

 

Fig. 2 Real and imaginary parts of the dynamic conductivity for DSFs under different Fermi levels in units ${\text{e}}^2/\hslash$ as a function of the normalized frequency $\hslash \text{ω}/{\text{E}}_{\text{F}}$. The parameters are set as $\text{ g}=40,\text{ and μ}=3\times {10}^4{\text{cm}}^2{\text{V}}^{-1}{\text{S}}^{-1}$ (the intrinsic time $\text{ τ}=4.5\times {10}^{-13}\text{s}$).

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3. Results and discussions

To illustrate the PIT effect, we numerically calculate the transmission spectra of the unit cell with two parallel DSF strips (${\text{L}}_1=40 \text{μm},{\text{L}}_2=32 \text{μm}, \text{s}=10 \text{μm}$), which is indicated by the red line in Fig. 3(a). For comparison, we also calculate the transmission spectra of the parallel structures alone with ${\text{L}}_1=40 \text{μm }$ and ${\text{L}}_2=32 \text{μm }$ respectively, as indicated by the blue and green line in Fig. 3(a). This implies that each of the two strips with specific geometric parameters has a resonant mode. It can be observed clearly that two transmission dips of the PIT spectrum are close their initial resonant mode. And the transmission peak within two resonant modes of the transmission spectra is activated through the coupling between two parallel DSF strips with unequal lengths. The Fermi energy of 70 meV is unchanged in all simulations for Fig. 3(a).

 

Fig. 3 (a) The PIT spectrum (${\text{L}}_1=40 \text{μm}$, $\text{s}=10 \text{μm}$) and simulated transmission spectra of the upper strip with ${\text{L}}_1=40 \text{μm}$ and lower strip with ${\text{L}}_2=32 \text{μm}$.The distribution of z-component of electric field at frequencies of (b) 1.434 THz, (c) 1.536 THz and (d) 1.626 THz corresponding to the PIT spectra.

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To get a further insight into the physical mechanism behind PIT effect in the simulated design, the distribution of z-component of electric field corresponding to the two transmission dips (1.434 and 1.626 THz) and transmission peak (1.536 THz) is investigated, as shown in the Fig. 3(b)-3(c), respectively. Two DSF strips can be regarded as optical dipole antennas and strongly excited by the incident light, and the excited optical dipole antennas interact with each other. When frequency is 1.434 THz, only the upper strip is excited strongly by the incident light, while the lower strip is excited weakly, as shown in Fig. 3(b). However, when frequency is 1.626 THz, only the lower strip is excited strongly by the incident light and the upper one is excited very weakly, as shown in Fig. 3(d). As for plasmon resonance at 1.536 THz, we observe that both of the two strips are excited simultaneously owing to the plasmon coupling between the two bright modes, which are similar to quadrupole antennas, as shown in Fig. 3(c).

To get more information about the PIT phenomenon, we numerically study the transmission spectra of the structure with different lateral displacements s (${\text{L}}_1=40 \text{μm}$, ${\text{E}}_{\text{F}}=70\text{meV}$), as shown in Fig. 4. From the Fig. 4(a), we can see that a low-energy peak and a high-energy peak are obtained when s = 0. With s increasing, the low-energy peak is broadened, while the high-energy one becomes narrower, and the width of transmission peak becomes smaller and the strength weakens gradually. It is mainly because the gradually weakened hybridization between the two resonant modes with the displacement s increases.

 

Fig. 4 (a) Transmission spectra at various lateral displacement s from 0 to $10 \text{μm}$, when L₁ = 40 um. (b),(c) The distribution of z-component of electric field at frequencies of 1.338 and 1.692 THz, when s = 0, respectively.

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The distribution of z-component of electric field corresponding to two transmission dips is investigated as shown in Fig. 4(b), 4(c) at frequencies of 1.527 and 1.626 THz, when s = 0, respectively. From Fig. 4(b), 4(c), it can be clearly found that the low-energy (1.527 THz) mode exhibits an out-phase resonance while the high-energy (1.626 THz) mode exhibits an in-phase resonance, which are analogous to the bonding and the anti-bonding modes in a hybridized-molecular system [33]. Since the phase resonances in two DSF strips are opposite, the bonding mode is similar to the quadrupole mode, so the bandwidth is narrow, which can be clearly found from Fig. 4(a). On the contrary, the phase resonances in two DSF strips are the same and the bandwidth of anti-bonding mode is broad. Meanwhile, the coupling strength between the two DSF strips weakens with the increases in s.

Then, we numerically study the transmission spectrum of the DSF strips with different ${\text{L}}_1$ ($\text{s}=10 \text{μm}$, ${\text{E}}_F=70\text{meV}$). As shown in Fig. 5, it presents the simulation transmission spectrum for different lengths ${\text{L}}_1$ from $32to4\text{4}\text{μm}$ with an increment of $\text{4}\text{μm}$ It can be clearly found that there is only one dip about the red line when ${\text{ L}}_1={\text{L}}_2=32 \text{μm}$, this is because the two strips have same resonant mode and phase. Once the state of equal lengths between the two strips is broken, the initial resonant dip gradually splits into two transmission dips, and then the PIT window appears. Obviously, the high-energy peaks are concentrated upon one point approximately at 1.62 THz, this is because the high-energy resonances are generated by the lower DSF strip whose length is fixed at ${\text{L}}_2=32 \text{μm }$ in our works, while the low-energy peak generated by the upper strip has a red shift with ${\text{L}}_1$ increases. Meanwhile, the strength and width of the transmission peaks heighten with the length ${\text{L}}_1$ increases. Compared with the curves of ${\text{L}}_1=40$ and ${\text{L}}_1=44 \text{μm}$, we find that two transmission dips are asymmetrical when ${\text{L}}_1=36 \text{μm}$, as shown by the green line. It is mainly because the phase and resonant mode of the two strips are very close, and they cannot split well. So, this kind of asymmetry will be more pronounced when ${\text{L}}_1$ is between $32$ and $36 \text{μm}$, and we can also understand like this: when the value of lateral displacement $\text{s}(\text{s}=10 \text{μm})$ is fixed, as ${\text{L}}_1$ decreases, the distance between the central axis of the two strips decrease, hence, the coupling strength enhances, then a low-energy peak and a high-energy peak appear in the transmission spectra.

 

Fig. 5 Transmission spectrum of the DSF strips with different lengths ${\text{L}}_1$ from 32 to $\text{44} \text{μm}$ with an increment of $\text{4} \text{μm}$ when $\text{s}=10 \text{μm}$,${\text{ E}}_{\text{F}}=70\text{meV}$.

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Since the DSFs are easier to process and more stable than graphene layers, we then numerically study the transmission spectrum at various thicknesses t_d of the DSF strips from 0.16 to $0.\text{22}\text{μm}$ with an increment $0.\text{02}\text{μm}$ as shown in Fig. 6(a). It is clear that the transmission spectrum exhibits a blue shift with the increase in t_d. Simultaneously, as t_d increases, all of the transmission values of dips gradually increase, while the transmission peaks show a linear decreasing trend. Besides, as mentioned before, the optical properties of the 3D Dirac semimetals can be substantially modified since the conductivity of the material can be changed by varying the Fermi energy. By tuning the Fermi energy, the electron density in DSFs is changed, leading to the change of resonant frequencies of DSFs plasmonic structures. To illustrate the tunability, both the transmission and absorption spectra are investigated under different Fermi energies when $\text{s}=10 \text{μm}, {\text{L}}_1=40 \text{μm}$, as plotted in Fig. 6(b). It is obvious that by a small change in the Fermi energy, the PIT effect can be easily tuned in the THz frequency regime and the resonant frequency is gradually blue-shift with the Fermi energy increases. Also, it is noteworthy that the transmission peak gradually decreases with the Fermi energy increases, while the corresponding absorption resonant peak gradually increases. This method of tunability is different from that used in metal-based systems [34, 35] where platforms with similar structures are investigated and changes are needed in the structures for tuning the transparency window of the EIT-like system. Also, the graphene-based PIT system [36] with similar structures provides and evaluates a tunable method, whereas DSFs have more excellent characteristics than graphene, as described above. Especially, the significant tunability can be achieved by changing the thickness of DSFs, which has been shown in Fig. 6(a).

 

Fig. 6 Transmission spectra (a) at various t_d with an increment $0.\text{02}\text{μm}$ and (b) transmission and absorption spectra under different Fermi energies (${\text{L}}_1=40 \mu \text{m},{\text{L}}_2=32 \mu \text{m},\text{s}=10 \mu \text{m}$).

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EIT response has been approved to have the capability to greatly slow down the speed of light [37, 38] due to the strong dispersion of transmission phase. It implies that the traversing time will be increased when a light pulse passes through the metamaterial structures. The slow-light effect attracts much interest because of its potential applications, such as buffers and light storages, especially for the tunable slow-light devices. The PIT phenomena demonstrated in Dirac semimetals plasmonics in our design offers a capability for electrical tuning. In our paper, dephasing times (ps) for the transmission resonant modes with different Fermi energies are quantified accounting to [39]. Corresponding values of about 8.69, 8.73 and 7.69 ps are obtained respectively. Simultaneously, Fig. 7 gives the transmission phase (°) and the corresponding group delay (ps) under different Fermi energies of DSFs. Group delay $t_g$ is calculated by $t_g=-\frac{d\phi \left(\omega \right)}{d\omega },$ $\text{ω}=2\text{π}f$ [40], where $\text{ω }$ is the angular frequency and $\text{φ}(\text{ω})$ is the transmission phase. Positive and negative group delays correspond to slow and fast light, respectively. As shown in Fig. 7(b), large positive group delays indicating slow light of more than 1.86 ps under different Fermi energies in the vicinity of the PIT transparency peak are obtained. Moreover, we can achieve the capability of tuning on and off the group delay by changing the Fermi energy.

 

Fig. 7 (a) Transmission phase (°) under different Fermi energies of DSFs, (b) group delay (ps) under different Fermi energies of DSFs

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4. Conclusion

In this work, we have successfully proposed that the PIT effect can be realized in DSFs known as “3D graphene” through bright-bright modes coupling in the THz regimes. Two parallel DSF strips couple with each other and the PIT window appears due to the weak hybridization. The properties of PIT system can be influenced by varying the lateral displacement between the two DSF strips and the length of the upper strip. In particular, the PIT window can be dynamically tuned by adjusting the Fermi energy of the DSFs, instead of refabricating the structures. At last, group delays under different Fermi energies are calculated in our proposed structure. All these features may find use in switches, sensors, and slow-light devices in the THz frequency regime.