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Abstract
By utilizing the three-dimensional Dirac semimetal (DSM)-strontium titanate (SrTiO3, STO) elliptical hybrid metamaterials, the tunable Fano resonances were systematically analyzed in the THz regime, for example, the effects of asymmetric degrees, DSM Fermi levels, and operation frequencies. Interestingly, an obvious Fano peak is observed by introducing a displacement (asymmetric degree) between STO and DSM resonators. In particular, the amplitude modulation depth (MD) of the Fano transmission peak (reflection dip) is 49.5% (86.65%) when the asymmetric degree ranges from 0 to 20 µm. Furthermore, on the condition that the asymmetric degree is larger than 10 µm, the LC resonance is also excited with an extraordinary Q–factor of more than 25. Additionally, by modifying the Fermi level of DSM layer, the amplitude MD of Fano transmission peak (reflection dip) is 32.86% (67.26%). The results facilitate our understanding of the tunable mechanisms of DSM metamaterials and potentially promote the development of novel plasmonic devices, including filters, modulators and sensors.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the continuous and accelerated development of terahertz (THz) technology, THz-based radiation source and detector, have been widely and practically applied in various fields of high-throughput wireless communication, (e.g. 6 G wireless communication systems), state-of-the-art biomedical sensors for imaging and diagnosis, and observational astronomy, etc. [1–5]. By utilizing the well-designed unit cells (meta-molecular) in a periodic manner, electromagnetic metamaterials (MMs) can be fabricated using photolithography as well as novel additive manufacturing techniques, based on the geometric configurations of subwavelength building blocks interacting with THz waves [6–8]. Besides, dielectric materials, metal and semiconductors with high permittivity can also be employed for the fabrication of MMs, which effectively surmounts and mitigate the inevitable Ohmic and radiation losses. All-dielectric materials (ADMs) enjoy the advantages of low dissipation, simple geometric structure, and relatively easy fabrication technique stage, platform for THz devices with high efficiency [9,10]. Thus, a proper low-loss and high-permittivity dielectric material is key to THz ADMs. Inheriting the merits of large electro-optic coefficients, strontium titanate (SrTiO3, STO), with both low dielectric loss and large permittivity, is a great candidate for fabricating ADMs structures [11–13]. With a strong polar soft mode, the permittivity of STO layer can be tuned by simply varying temperatures or bias electric fields [14,15]. Some research have been carried out to investigate STO MM structures.
Generally, the electromagnetic responses of ADMs are fixed once the materials and geometrical parameters are chosen. As an important topological semimetal and peculiar quantum state of matter, three-dimensional (3D) Dirac semimetal (DSM) is also referred to as “3D graphene” and has attracted tremendous attention, which can serve as an active medium to solve this dilemma [16–21]. In addition to the similar merits to graphene (e.g. strong mode confinement and excellent tunable properties), 3D DSM also possesses several advantages, e.g. high mobility and Fermi velocity [22–29], which in turn enhances its tunability and plasmonic properties. The bulk properties of 3D DSM also help break the limitation of thickness, and contribute to more degrees of freedom in the design of functional devices [30–38]. In particular, with an ultrathin film of Cd3As2 (100 nm), the broadband modulation of THz waves was experimentally investigated using the active photoconductivity method. The results demonstrated that the electron mobility was about 7200 cm2 (V·s)-1 and the momentum scattering time was 0.157 ps, which were comparable to the kinetic inductance superconductors. The 3D DSM supported THz striped patterned and asymmetric split-ring resonators also showed obvious dipolar and Fano resonances, and the resonant frequency shift reached more than 86 GHz by changing the optical pump light configurations [39]. Additionally, using a hybrid structure of rose-shaped DSM and STO layers, a bi-tunable THz MM absorber was fabricated, for which the peak absorptivity was modulated in the range of 70%–99%, when the Fermi level was adjusted from 10 to 80 meV. When the temperature increased from 200 K to 300 K, the absorption frequency was tuned from 2.66 to 3.69 THz with a nearly fixed absorption amplitude [40].
Indeed, achieving efficient control of THz waves with simple MM structures and satisfactory performance is in urgent need to develop high-sensitive sensors and high–speed wireless communication systems [41–43]. The modified microstructures significantly affect the resonant curves, and can be employed to modulate the resonant curves. In order to obtain flexible resonant curves with high Q–factor, the major propagation properties of the modified DSM-STO hybrid MMs (i.e. the DSM resonator shifts the STO layer with a displacement δ along the x direction) are determined in the THz regime, for example, the effects of asymmetric degree, Fermi levels, and temperature. The findings demonstrate that a sharp Fano resonance peak is achieved, and the amplitude MD reaches more than 49.5% when the asymmetric degree varies from 0 to 20 µm. Furthermore, an obvious low-frequency LC resonance is also excited if the asymmetric degree is larger than 10 µm with a Q–factor of >25.
2. Research methods
Figures 1 illustrates the geometric configurations of elliptical DSM-STO hybrid resonators MMs structure. The thickness of the polyimide layer is 2 µm. The polarization of the incident THz waves is along the y direction.
Fig. 1. Geometric configurations of the elliptical DSM-STO hybrid MM structures, which are positioned at the SiO2-Si-polyimide multilayers. (a) The side-view and (b) Top-view of the proposed hybrid MM structures. The incident THz wave usually transmits through the MM structures along the z direction. The bias voltage is applied between the DSM and doped Si layer to change the Fermi level. The thicknesses of SiO2 and Si layers are 0.03 µm and 2 µm, respectively.
The frequency-dependent dielectric constant of STO layers within the THz range is determined as follows [44]:
The complex conductivity of 3D DSM under the framework of Kubo formalism is described as follows [45,46]:
3. Results and discussion
The DSM layer exhibits great tunable properties, and the strontium titanate (SrTiO3) displays large permittivity, low dissipation losses, and strong mode confinement. Thus, the elliptical DSM-STO (DS) hybrid resonators were utilized to manipulate the THz waves. The key features of the proposed elliptical hybrid MM structure were systematically analyzed with CST Microwave Studio software using the finite integration approach. In the simulation, the magnetic boundary condition and electric boundary condition are adopted along the x and y directions, and in the z direction open (add space) boundary condition is utilized. From the obtained S-parameters, the transmission, reflection and absorption curves are defined by T=|S21|2, R=|S11|2, and A = 1−T−R. The resonant curves of elliptical DS hybrid resonators at various asymmetric degrees are shown in Figs. 2(a)–2(c), respectively. They are introduced by shifting the DSM layer by a displacement δ along the x direction, as given in Fig. 1(b). If the asymmetric degree is zero, i.e. for the symmetric structure, only broad dipolar resonance is excited, the reflection dominates, and the transmission curve shows a broad resonant dip at 0.9702 THz. However, if the asymmetric degree of DSM layer δ is nonzero, a sharp obvious peak appears, which may be attributed to the Fano resonance. The Fano resonant peak amplitude increases with an increasing asymmetric degree δ. When the values of δ are 3, 10 and 20 µm, the respective Fano peak amplitudes are 0.3722, 0.6517 and 0.7741, respectively, and the corresponding amplitude is 49.5%. Figure 2(b) displays the reflection curves with an obvious reflection dip. With increasing asymmetric degree, the reflection dip decreases significantly. When the values of δ are 3, 10 and 20 µm, the dip values of Fano resonance are 0.3425, 8.259 × 10−2, and 4.573 × 10−2, respectively. The coinciding amplitude MD of the reflection dip is 86.65%. For the case of dissipation, if the asymmetric degree is small, the absorption is large and increases by approximately 50% when δ is 5 µm. However, if the asymmetric degree increases further, the Fano resonance becomes stronger, and the absorption decreases. Thus, on the condition that the asymmetric degree is small, the absorption and reflection dominate, and the transmission Fano peak is small. However, at large asymmetric degree, where the Fano resonance is strong, the transmission increases, and the reflection and absorption decrease. The Q-factor describes the underdamped property of a resonator and is defined as the ratio of resonant frequency fres to the full width at a half maximum (FWHM), i.e. Q–factor = fres/FWHM. Additionally, to judge the overall performances of the curves, the figure of merit (FOM) is given by FOM = Q–factor × Am, in which Am is the resonant strength. The Q−factor and FOM of the transmission Fano peak are demonstrated in Fig. 2(d). If the asymmetric degree is small, the resonant peak is sharp with a large Q−factor, which reaches >20. As the asymmetric degree increases, the absorption increases significantly, resulting in a rapid reduction in the Q-factor. Simultaneously, the resonant strength increases with asymmetric degree, and thus the FOM reaches a peak if the asymmetric degree is about 10 µm. Additionally, at low frequency, near 0.4 THz, the LC resonance is also excited. With increasing asymmetric degree, the transmission and reflection dip decrease, but the absorption increases. As asymmetric degree increases, the LC resonant frequency is modulated in range 0.4857−0.3812 THz. Owing to the relative large thickness of DSM layer, the Q−factor is higher (approximately 18−25), which is distinctively different from the thin layer of graphene membrane.
Fig. 2. (a) Transmission, (b) reflection, and (c) absorption curves of the proposed DS hybrid structure at different displacements. (d) The Q–factor and figure of merit of transmission curves versus asymmetric degrees. The major and minor semi-axes of elliptical DS MMs are 66 µm and 20 µm, respectively. The period lengths along the x and y directions are both 140 µm.
2D simulation results are also good means to clarify the propagation mechanism of elliptical DS hybrid MM structure. Figure 3 displays the surface current density (SCD) and magnetic fields (MFs) at the LC, low frequency, and Fano resonances. These corresponds to the A, B, and C points in Fig. 2, at the frequencies of 0.4325, 0.9626 and 1.276 THz, respectively. As shown in Fig. 3(a), the surface currents form a circular loop, a typical LC resonance, in the interfacial area of DSM and STO layers, which is also confirmed by the field distributions in Fig. 3(d). This may be attributed to the fact that the transmission dip at point A is due to LC resonance. For the point B, it is a typical dipolar resonance, where the surface current flows along the y direction. Figure 3(d) also shows that the MFs are distributed over the whole hybrid resonators. The resonant point C for the asymmetric resonant peak arises from the Fano resonance, and the SCD and MFs can be found in Figs. 3(e) and 3(f), respectively. For the Fano resonance, a symmetry dipolar mode strongly interacts with incident waves, which is known as the bright mode, a highly symmetry mode weakly interacts with the incident wave, which is known as the dark mode. Due to the displacements of DSM and STO layers, the surface currents flow in different directions at the middle, upper and lower sections of the elliptical resonators, as displayed in Fig. 3(e). Since the surface currents oscillate out-of-phase, a net current and dipole moment appears, which leads to an obvious Fano resonant peak.
Fig. 3. (a)-(c) The surface current densities and (d)-(f) magnetic fields for the DSM-STO elliptical MM structures at LC, low frequency resonance, and Fano resonance. The corresponding resonant frequencies are 0.4325, 0.9626, and 1.276 THz, respectively. The asymmetric degree is 15 µm. The major and minor semi-axes of elliptical DS MMs are 66 µm and 20 µm, respectively. The period lengths along the x and y directions are both 140 µm. The thicknesses of STO and DSM layers are both 2 µm.
The resonant curves of DSM-STO hybrid resonators at various Fermi levels are displayed in Figs. 4(a)–4(c). At small Fermi levels, the Fano resonance is weak, for example, Ef = 0.01 eV, and the peak value is about 0.4948. With increasing Fermi level, the DSM layer demonstrates greater plasmonic properties, where the Fano resonance increases, and the resonant frequency also displays a blue shift. Particularly, at Fermi levels of 0.01, 0.05, 0.10, and 0.15 eV, the dielectric properties of DSM layers are −8.530 × 101 + 3.789 × 101i, −2.401 × 103 + 8.569 × 102i, −9.638 × 103+ 3.417 × 103i, and −2.170 × 104 + 7.683 × 103i, respectively. Accordingly, the amplitudes of Fano resonances are 0.4948, 0.6283, 0.7172 and 0.737, respectively. The corresponding amplitude MD is 32.86%. The influence of Fermi levels on the reflection curves is demonstrated in Fig. 4(b), where the reflection dip decreases dramatically. For example, at the Fermi levels of 0.01, 0.05, and 0.10 eV, the respective peak values are 0.1729, 0.07707, and 0.05972, and the coinciding amplitude MD is 67.26%. The impact of Fermi level on the dissipation is shown in Fig. 4(c). The resonant strength is not very large at a small Fermi level, and the absorption is large; while at higher Fermi levels, the reflection increases, and the effect of absorption reduces. These phenomena indicate that at small Fermi level, the absorption dominates; but at larger Fermi level, the DSM layer exhibits greater plasmonic properties, the resonant strength increases, and the absorption reduces. Figure 4(d) illustrates the simulation results of Q−factor and FOM. Since the DSM layer manifests good metal properties, the Fano resonance is strong with the Q−factor about 10. Meanwhile, the amplitude of Fano resonant peak increases with Fermi level, resulting in FOM enhancement. Additionally, it can be found from Fig. 4 that, at low frequency, near 0.4 THz, the LC resonant is excited. With increasing Fermi level, the transmission and reflection dip decrease, while the absorption increases, and the LC resonant frequency (dip value) is modulated in the range of 0.3698-0.4325 THz (0.4067-0.3009), frequency (amplitude) MD is about 14.5% (26.01%).
Fig. 4. (a) Transmission, (b) reflection, and (c) absorption curves of the proposed DS hybrid structure at different Fermi levels. (d) The Q−factor and FOM of transmission curves versus Fermi levels (0.01, 0.02, 0.05, 0.08, 0.10, and 0.15 eV). The displacement between DSM and STO layers is 15 µm.
The effects of different temperatures on the resonant curves of elliptical DS hybrid MM structures are shown in Figs. 5(a)–5(c). At relatively low temperature, the STO permittivity is large, the displacement current play an important role. As a result, the Fano resonant is strong, e.g. the peak value is about 0.737, if temperature is 70 K. As temperature increases, the STO layer provides worse dielectric properties, the displacement current reduces, and the interaction between hybrid MM structure becomes weaker. Consequently, the peak value of Fano resonance decreases with temperature, and the resonant frequency also illustrates a blue shift. In particular, at the frequency of 1.25 THz and the temperatures of 70, 150 and, 300 K, the STO permittivity are -2.541 × 103 + 3.962 × 102i, 1.331 × 103 + 3.562 × 102i, and 3.653 × 102 + 5.865 × 101i, respectively. The peak values of Fano resonances are 0.737, 0.3628 and 0.5687, and the amplitude MDs is 66.59%. The influence of temperatures on the reflection curves is demonstrated in Fig. 5(b). As temperature increases, the reflection dip value increases significantly. When the temperatures are 70, 150 and 200 K, the peak values (resonant frequencies) of Fano resonances are 0.05661, 0.2136 and 0.3631 at 1.280, 1.293 and 1.303 THz, respectively. The corresponding amplitude MD is 84.41%. For the case of loss, if the temperature is small, e.g. < 150 K, the resonant strength is not very large, and the absorption is large, which reaches more than 40% when temperature is 150 K and corresponds with the reflection dip. At higher temperature, the Fano resonance becomes stronger, the absorption and reflection decrease, and the transmission increases. On the condition that the temperature is small, the transmission dominates. However, with increasing temperature, the STO layer dielectric constant reduces, the transmission decreases, and the reflection and absorption increase. The Q–factor of transmission peak are shown in Fig. 5(d). Since the STO layer manifests good dielectric properties, the Fano resonances are strong, its Q–factor reaches more than 12. With increasing temperature, the resonant strength and FOM decrease, as shown in Fig. 5(d). In addition, at low frequency, the LC resonance is excited, which manifests a large Q–factor of >20. As temperature increases, the transmission and reflection dip decrease, but the absorption increases. With increasing Fermi level, the LC resonant frequency (dip amplitude) of transmission curves is modulated in the scope of 0.4325-0.8144 THz (0.0026-0.3009), and the frequency (dip amplitude) MD is about 46.89% (99.14%).
Fig. 5. (a) Transmission, (b) reflection, and (c) absorption curves of the proposed DS hybrid structure at different temperatures. (d) The Q−factor and FOM versus temperatures. The Fermi level of DSM layer is 0.15 eV. The major and minor semi-axes of elliptical MMs are 66 µm and 20 µm, respectively. The pitch lengths along the x and y directions are both 140 µm. The thicknesses of STO and DSM layers are both 2 µm. The asymmetric degree is 15 µm.
The SCD and MFs of DSM-STO hybrid resonators at various STO temperatures are shown in Fig. 6. The results show that temperature significantly affect LC resonance. As the temperature increases, the interaction of DSM and STO layers increases, thereby enhancing the surface current and fields distribution. Additionally, with increasing temperature, the resonant frequency indicates a blue shift. This is elucidated by the following reasons. An elliptical resonator is known as a small LC circuit, in which the resonant frequency (fres) can be measured by the effective inductance and effective capacitance L and C, i.e. ${f_{\textrm{res}}} \propto 1/\sqrt L /\sqrt {{\varepsilon _0}{\varepsilon _r}S/l} $. The effective capacitance C is closely associated with the temperature dependent STO permittivity. Consequently, as the decrease of temperature, the capacitance reduces, causing a blue shift of resonant curves. At 70, 150 and 300 K, the STO permittivity are -7.614 × 103 + 3.276 × 103i, 1.331 × 103 + 3.562 × 102i, and 3.372 × 102 + 3.931 × 101i, respectively. Figures 6(g)–6(k) are the simulation results of Fano resonances. As the temperature increases, the interaction of DSM and STO layers decreases, thereby reducing the surface current and field distribution. In this case, the effect of Fano resonance is improved, and the absorption increases.
Fig. 6. (a)-(c), (g)-(i) Surface current density and (d-f), (j-k) magnetic fields of the elliptical DSM-STO MM structures at LC, and Fano resonances, respectively. The temperatures of DSM layers are 70, 150, and 300 K, respectively. Accordingly, for the LC resonance, the resonant frequencies are 0.4325, 0.7194, and 0.8144 THz. For the Fano resonances, the resonant frequencies are 1.276, 1.306 and 1.426 THz. The period lengths along the x and y directions are 140 µm. The thicknesses of STO and DSM layers are both 2 µm.
4. Conclusion
The tunable propagation properties of the DSM-STO hybrid MM structures were comprehensively analyzed in the THz regime, for example, the influences of the displacement between hybrid resonators, DSM Fermi levels, and temperatures. These findings demonstrate that by introducing a displacement between STO and DSM resonators, an obvious Fano peak is observed. When the asymmetric degree ranges from 0 to 30 µm, the amplitude MD of transmission peak (reflection dip) is 49.5% (86.65%). Furthermore, if the displacement is more than 10 µm, the low-frequency LC resonance is also excited with a Q−factor larger than 25. By varying the Fermi levels, the resonant curve was adjusted within a broad range, the amplitude MDs of transmission peak (reflection dip) is 32.86% (67.26%). This study can enhance our understanding of the tunable mechanisms of DSM MMs and develop more efficient functional devices, including filters, modulators, and sensors.
Funding
National Natural Science Foundation of China (62205204, 61674106); Shanghai Local College Capacity Building Project (22010503300); Natural Science Foundation of Shanghai (21ZR1446500); Funding of Shanghai Normal University (SK202240).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the corresponding authors upon reasonable request.
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