3D Dirac semimetals supported tunable terahertz BIC metamaterials

: Based on the 3D Dirac semimetals (DSM) supported tilted double elliptical resonators, the tunable propagation properties of quasi-bound in continuum (BIC) resonance have been investigated in the THz regime, including the effects of rotation angles, DSM Fermi level, and the configuration of resonators. The results manifest that by altering the rotation angle of elliptical resonator, an obvious sharp BIC transmission dip is observed with the Q -factor of more than 60. The DSM Fermi level affects the BIC resonance significantly, a sharp resonant dip is observedifFermilevelislargerthan0.05 eV,resultingfrom the contributions of reflection and absorption. If Fermi level changes in the range of 0.01–0.15 eV, the amplitude and frequency modulation depths are 92.75 and 44.99%, respectively. Additionally, with the modified configurations of elliptical resonators, e.g. inserting a dielectric hole into the elliptical resonator, another transmission dip resonance is excited and indicates a red shift with the increase of the permittivity of the dielectric filling material. The results are very helpful to understand the mechanisms of DSM plasmonic structures and develop novel tunable THz devices, such as modulators, filters, and sensors in the future.


Introduction
In recent years, terahertz (THz) waves witness the innovative applications in the fields of astronomical observation, medical imaging, homeland security identification, and high-speed 6G wireless communication [1][2][3][4][5][6]. For instance, broadband THz emission via surface optical rectification from a 19 nm thin layer of indium tin oxide was demonstrated, due to the focus of the pump laser fields related with the epsilon-near-zero effect, the generated THz signal was enhanced significantly and overcame the restriction of the phase-matching condition, its bandwidth was over 3 THz [4]. But to the further substantial developments of THz science and technology, it is highly demanded to explore functional devices and components with fine performances. Artificially made of well-ordered subwavelength resonators, metasurfaces are capable of effectively regulating the electro-magnetic properties of incident waves [7][8][9][10][11][12]. However, the further exploration of metasurfaces devices to control the light in an arbitrary desirable manner is hindered by the large radiation losses and poor electrical tunability. Novel emerging materials, such as black phosphorus, transition metal molybdenum disulfide, graphene, and topological materials, provide good platforms for the exploration flexible functional devices [13][14][15][16][17][18][19][20]. As an important type of topological materials, Dirac semimetals (DSM) manifests the merits of linear dispersion, large fermion degeneracy, and especially the dynamical manipulation of conductivity, which sheds new possibilities design of THz devices [21][22][23][24][25][26].
Originally proposed by von Neuman and Wigner in 1929, bound states in the continuum (BICs), i.e. a perfected non-radiative discrete bound state coexisted within a continuum spectrum of spatially extended states, is formed through destructive interference between leaky modes and shows an infinite Q-factor [27][28][29][30]. However, thanks to the fabrication imperfections, roughness, and material loss, a quasi-BIC resonance with sharp peak and finite Q-factor appears [31][32][33][34]. Inhibiting the peculiar ability of full suppression of radiation losses and strongly confined modes, the quasi-BICs in metasurfaces structures are closely associated with Fano resonances, trapped modes, and plasmon induced transparency [35][36][37][38], which attracts the attentions of many researchers. For instance, by utilizing pairs of tilted Si nano-bars F. Yesilkoy et al. investigated the quasi-BIC phenomenon in the near-IR spectral region, the Q-factor was about 144, near-field enhancement was increased by 40 times, and the refractometric sensitivity reached about 263 nm per refractive index unit [39]. Based on a hybrid structure of uniform graphene membrane and Si nanodisks, X. Wang et al. showed due to the enhancing interaction between the radiation engineering and BIC, the absorption bandwidth was modulated more than two orders of magnitude, i.e. 0.9-94 nm, by changing the asymmetric parameter of metasurfaces, the Fermi level, and layer number of graphene [40]. By covering part section of metal split ring metasurface with a thin Ge layer of thickness about 500 nm, a dynamically controllable quasi-BIC resonance was excited in the THz region, 200% transmission intensity modulation of the quasi-BIC resonance was achieved by photo-excited the Ge stripe, and the recovery time was within 7 ps [41]. With a thin MgF 2 layer inserting into periodic Si nano-pillars arranged in square lattices and Ag substrate, a hybrid dielectricmetal supporting symmetry protected Friedrich-Wintgen BIC resonance was demonstrated, which increased the lifetime of optical mode and minimized the mode volume simultaneously, the electric field was strongly confined in the dielectric particles and reduced the mode volume one order of magnitude [42].
It is crucial to develop flexible, low-cost, and high efficient THz devices with simple fabrication methods. The performances of MMs structure are closely associated with the configuration of unit cells, such as tilted resonators, inserting a hole in the resonator or adopting the hybrid structures [43,44]. For noble metal MMs resonators, the Q-factor of resonant curve is not very large, and the operating wavelength is designed at a fixed value. Similar to graphene layer, 3D DSM layer inhibits strong light confinement, low dissipation, and good tunable conductivity. Furthermore, 3D DSM has also several advantages, such as the higher Fermi velocity and mobility, surmounting the restriction of thickness and an additional structural degree-of-freedoms in the construction of functional devices [17][18][19][20][44][45][46]. It is widely expected DSM is good platform to design novel flexible functional devices. To explore high efficient tunable THz devices, the tilted elliptical DSM MMs have been investigated, indicating an obvious sharp BIC transmission dip with the Q-factor of more than 50. The DSM Fermi level affects the resonant curve significantly, sharp resonant curve is achieved if Fermi level is larger than 0.05 eV, the amplitude and frequency modulation depth (MD) are 92.75 and 44.99%, respectively. Additionally, with the modified configuration of elliptical resonators (such as the hetero-structure hybrid resonators or dielectric filling materials in the DSM resonators), another transmission resonant dip is observed. Under the framework of Kubo formalism in random phase approximation, the longitudinal complex dynamic conductivity of the 3D DSM can be expressed as [47]:

Structural design and research methods
in which G(E) = n(−E) − n(E), n(E) is the Fermi distribution function, E F indicates the Fermi level, k F denotes the Fermi wave-vector, = ℏ /E F , k F = E F /ℏv F represents the Fermi momentum, v F is Fermi velocity, E c remarks the cutoff energy beyond which the Dirac spectrum is no longer linear, g is the degeneracy factor. The permittivity of 3D Dirac semimetals can be obtained using the following formula, where b is the effective background dielectric ( b = 1, g = 40, for AlCuFe quasi-crystals), 0 is the permittivity of vacuum.
The Q-factor means the rate of the stored energy and the energy loss in the resonator, which can be expressed as: FWHM is the full width at a half maximum of the resonance peak.
To measure the trade-off between Q-factor and resonant strength, the figure of merits is defined as following, in which A m as the amplitude strength of the resonant curve.

Results and discussion
We study the BIC resonances based on a symmetry protected quasi-BIC system composed of pairs of the tilted elliptical DSM resonators, as given in Figure 2(a). The orientation of each elliptical resonator is characterized by a rotation angle between the y axis and the long axis of the elliptical bars, thus the asymmetric parameter is defined as sin . Figures 2(b)-(d) show the transmission, reflection, and absorption curves at different tilted angles. If the polarization is along the y direction, the elliptical resonator excites an obvious transmission resonant dip; the reflection curve also indicates an obvious peak, as the violet line given in Figure 2(b), which results from the dipolar resonance of DSM resonators in the THz region. However, if the polarization is along the x direction, on the condition that the tilted angle is zero degree, i.e. for the symmetric structure, this proposed DSM elliptical structure supports a symmetry-protected BIC and cannot couple to the free space radiation due to symmetry protection. In this case, the DSM resonator cannot excite obvious transmission dip because the semi-axis length along the x direction a x is small, i.e. the black line in the Figure 2(b). If the tilted angle is not zero, a small transmission resonant dip appears. As the tilted angle increases, the resonant strength of transmission curve becomes stronger, the electric inductance increases. Since the resonant frequency are proportional to the 1/(LC) 1/2 , the according transmission respectively. Thus, if the tilted angle changes in the range of 2-30 • , the amplitude and frequency modulation depths are 99.56 and 33.36%, respectively. The influences of rotation angles on the reflections curves can be found in Figure 2(c). If the value of is smaller than 5 • , the reflection curve peak is not very large. As the rotation angle increases, the BIC phenomenon becomes stronger, the reflection curve indicates a strong peak and obvious red shift, which corresponds with the transmission dip in Figure 2 48 and 33.58%, respectively. The reasons are given in the following. When the tilted angle is small, the overall radiative loss is suppressed significantly; a small sharp absorption peak appears, as the red and green lines given in Figure 2(d). As the tilted angle and asymmetric parameter increase, the gap distance between the two elliptical resonator decreases, the interaction between them increases. Thus, much more modes couple into free space, the dissipation increases, resulting in a larger dissipation and broader spectral line width, i.e. the magenta and orange lines given in Figure 2. However, if the tilted angle increases further, larger than 10 • the elliptical bar length along the polarized x-direction increases significantly. The low lossy dipolar resonance plays a dominated role and leading into the dissipation reduction again, as the magenta and orange lines given in Figure 2(d). Therefore, the absorption curves show a peak value at certain tilted angle, about 10 • . In a word, at small tilted angle the absorption dominates, the resonant peak position is not sensitive to the tilted angle. On the other hand, if the tilted angle is larger than 10 • , the reflection takes an important role and results into sharp BIC transmission resonant dip. The Q-factor and FOM can be found in Figure 2(e). At small tilted angle, the overall radiative loss is suppressed significantly, the Q-factor is larger than 60. As the tilted angle increases, the Q-factor decreases. However, the resonant strength becomes stronger with increase of the value of tilted angle. Thus, the FOM shows a peak at certain angle, about 5-8 • , as given in Figure 2(e). To have a deep understanding of the tunable mechanisms on the propagation properties, the surface current density and magnetic fields (H z ) of elliptical MMs structures at different tilted angles have been demonstrated in Figure 3. The incident THz waves drive a surface current flowing along the elliptical BIC resonators, and the directions along the left and right resonators are opposite. Thus, a circular loop is formed on the condition that the tilted angle is large, which results in a charge accumulation at gap of the tilted elliptical resonators. The simulation results of magnetic fields can be found on Figure 3(d)-(f). The gap distance between the double DSM elliptical resonators is large at small tilted angle; the interaction between them is weak. As the tilted angle increases, the gap distance reduces, the interaction between resonators increases significantly, the resonant strength becomes stronger. Additionally, since the surface current increases obviously at large tilted angle, the electric inductance enhances as well, which results into the according transmission resonant dip manifests a red shift.
In analogy to two-dimensional graphene, the complex conductivity and permittivity of 3D DSM can also be dynamically modulated by an applied bias voltage, which can be utilized to manipulate resonant curves efficiently. Figure 4 shows the effects of Fermi levels on the resonant curves. At small Fermi level, e.g. E f < 0.02 eV, the tilted elliptical resonator can't excite obvious BIC transmission dip, as given in Figure 4 Figure 4(d). As Fermi level increases, DSM layer shows better plasmonic properties, the resonant strength of BIC phenomenon becomes stronger, the Q-factor and amplitude increases, which results into the FOM enhancing and reaches a peak value of about 20. Figure 5 shows the surface current density and magnetic fields of elliptical MMs structures at different Fermi levels. The polarization is along the x direction. The Fermi levels are 0.02 eV, 0.05 eV, and 0.10 eV. From Figure  5(a)-(c), we can find that the surface current flows along the left and right elliptical resonator along different directions. The quasi-BICs boost the electric field enhancement inside metasurfaces. As Fermi level increases, the high permittivity of DSM layer at high Fermi level results in a small skin depth and low material losses. Thus, the resonant strength becomes stronger, and the resonant curve becomes sharper at larger Fermi level. The simulation results of magnetic fields can be found on Figure 5(d)-(f). If the Fermi level is small, the interaction between the left and right elliptical unit cell is weak, the magnetic fields is not stronger. As Fermi level increases, the interaction between the BIC resonators increases because of the better plasmonic properties of DSM layer. If the Fermi level is larger, e.g. 0.10 eV, the interaction between the left and right elliptical resonators are very stronger, as given in Figure 5(f). Figure 6 shows the propagation properties of the DSM modified elliptical BIC structure, i.e. a dielectric hole is inserted into the subwavelength resonator, as given in the inset in Figure 6(b). The permittivity of the dielectric filling materials for air, Teflon, polyimide (PM), SiO 2 , Al 2 O 3 , and Si are 1.0, 2.1, 3.5, 3.9, 9.9, and 11.9, respectively [48,49]. Due to the broken symmetry, another resonant dip at high frequency appears which is significantly affected by the dielectric filling materials. The strength becomes weaker with the increase of the permittivity of dielectric filling material, and the resonant dip also indicates a red shift. For example, if the dielectric filling materials are air, Teflon, polyimide (PM), SiO 2 , and Si, the resonant dip amplitudes (frequencies) are 0.2578 (1.225 THz), 0.2778 (1.202 THz), 0.3035 (1.175 THz), and 0.4604 (1.042 THz), respectively. The reflection curves for the different dielectric filling materials can be found in Figure 6(b). By utilizing the dielectric filling materials with larger permittivity, the reflection peak reduces and the resonant frequency indicates a red shift. For example, when the dielectric filling materials are air, Teflon, SiO 2 , and Si, the resonant dip amplitudes (frequencies) are 0.3276 (1.202 THz), 0.3080 (1.179 THz), 0.2773 (1.145 THz), and 0.1569 (1.027 THz), respectively. The according amplitude (frequency) modulation depth is 52.11% (14.56%). The dissipation curves can be found in Figure 6(c). With the modified hollow elliptical resonator, an absorption peak at high frequency is excited. Furthermore, as the permittivity of dielectric filling material increases, the absorption peak at high frequency enhances and moves to low frequency. From above discussions, it can be found that the high frequency transmission dip is mainly associated with the dielectric materials, the reflection contribution is relatively small, but the dissipation plays an important role. The value of Q-factor and FOM can be found in Figure 6(d). As the permittivity of dielectric filling materials increases, the mode confinement improves, the interaction of dielectric filling material with THz waves increases, resulting into the Q-factor enhancing. However, since the resonant strength of high frequency resonance becomes smaller, the FOM decreases with the large permittivity of dielectric filling materials, as given in Figure 6(d).

Conclusions
By depositing the planar arrays of tilted DSM elliptical MMs patterns on the SiO 2 /Si layers, the tunable propagation properties of BIC resonance are given and discussed in the THz regime, taking into accounting the tilted angles, Fermi levels, operation frequencies, and different dielectric filling materials. The results manifest that an obvious sharp BIC transmission dip can be observed, the Q-factor and FOM reach more than 60 and 20, respectively. The amplitude and frequency modulation depths of BIC resonance reach more than 99.56 and 33.36% if the tilted angle changes in the scope of 2-30 • . The BIC resonant curves are closely associated with DSM Fermi level, the sharper transmission dip can be achieved at larger Fermi level, e.g. if Fermi level changes in the range of 0.01-0.15 eV, the amplitude and frequency MD are 92.75 and 44.99%, respectively. If Fermi level is small, the transmission dip results from the absorption, while the contribution of reflection plays an important role if Fermi level is larger than 0.05 eV. Additionally, by introducing a hole in the elliptical DSM structure, another transmission resonant dip is excited at high frequency, which becomes weaker and indicates a red shift with the increase of the permittivity of the dielectric filling material. The results are very helpful to understand the mechanisms of the DSM BIC plasmonic structure and develop novel tunable THz devices, such as modulators, filters, and sensors in the future.