## Abstract

In this paper, a bulk Dirac semimetals (BDSs) based tunable narrowband absorber at terahertz frequencies is proposed and it has the attractive property of being polarization-independent at normal incidence because of its 90° rotational symmetry. Numerical results show that the absorption bandwidth is about 1.469e-2 THz and the total quality factor *Q*, defined as *Q* = *f*_{0}/Δ*f*, reaches about 94.6, which can be attributed to the low power loss of the guided mode resonance in the dielectric layer. The simulation results are analyzed with coupled mode theory. Interestingly, on the premise of maintaining the absorbance at a level greater than 0.95, the absorption frequency can be tuned from 1.381 to 1.395 THz by varying the Fermi energy of BDSs from 50 to 80 meV. Our results may also provide potential applications in optical filter and bio-chemical sensing.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Perfect absorber, in principle, is obtained by minimizing the reflection and eliminating the transmission. Since a perfect absorber based on metamaterials was first reported in 2008 by Landy *et al* [1], metamaterials based absorbers have been extensively studied and demonstrated in much spectral range, including microwave [2], terahertz [3–5], infrared [6, 7] and visible regions [8–10]. According to their absorption bandwidth, they can be categorized into broadband absorbers and narrowband absorbers. The reason why they are so popular can be attributed to their potential in wide range of applications, while broadband absorbers are extensively be used in solar power harvesting and thermo-photovoltaics [11], narrowband absorbers can be used as sensors [12–14] and filters [15, 16], as well as thermal radiation tailoring [17, 18]. In general, an absorber is usually designed as a metal-dielectric-metal (MDM) configuration, where the above metal layer is used to implement the impedance matching and the below metal layer can be served as ‘Salisbury screen’ of the absorber to block the transmission if it is thick enough, and the MDM configuration will be served as an optical device for enhancing the transmission if the below layer is not thick enough [19]. The absorption bandwidths of these absorbers are relatively broad due to the radiative damping and nonradiative losses, which is good for a broadband absorber while hampers the applications of a narrowband absorber. Fortunately, nonradiative losses can by reduced by employing materials of low loss, such as dielectric [20, 21], to replace the above metal layer. Liu *et al* [22] designed a narrowband absorber with its bandwidth down to 2 nm in the visible range based on a triple-layer dielectric metamaterials structure coupled with a metal substrate, and the multispectral light perfect absorption can be attributed to optical cavity resonances and the plasmon-like dipolar resonance of the dielectric resonators and their hybridization effects.

Unfortunately, perfect absorbers based on traditional noble metals, such as gold, silver, or copper, are designed at a fixed absorption peaks and the geometric parameters of the absorber have to be carefully re-optimized if one wants to tune the absorption peaks to other frequency regions. Later, this problem was solved by introducing graphene to the absorber as the optical response of graphene is characterized by its surface conductivity *σ* [23–25] that greatly relates to its Fermi energy *E _{F}* which can be dynamically tuned by applying bias voltage [26]. Actively tunable graphene-based plasmonic devices have been widely studied, such as modulator [27, 28] and sensor [29, 30], etc. Recently, Liao

*et al*[31] proposed a tunable narrowband mid-infrared TE-polarization absorber, and the narrowband can be attributed to the low power loss of the guided mode resonance based on dielectric layer while the tunability of the proposed absorber is realized by introducing a graphene layer to the absorber. And Zhang

*et al*[32] proposed a novel method to achieve tunable frequency by utilizing multiple varactors instead of graphene. Most recently, a novel state of quantum matter—bulk Dirac semimetals arouses great interests among researchers which can be considered as “3D graphene” for reason that the permittivity functions of it can also be dynamically adjusted by changing its Fermi energy

*E*through alkaline surface doping [33, 34]. The BDSs manifests a metallic response at frequencies lower than Fermi energy while a dielectric response becomes dominated at frequencies higher than Fermi energy [35]. It may be more convenient than graphene if the BDSs be used to design a tunable absorber since the BDSs can be served as ‘Salisbury screen’ of an absorber to block the transmission thanks to its metallic property at the THz frequencies.

_{F}In this paper, a BDSs based tunable narrowband absorber at terahertz frequencies is proposed and it has the attractive property of being polarization-independent at normal incidence because of its 90° rotational symmetry. Different from reported metamaterials absorbers, this design not only combines new materials to achieve tunable frequency, but a simpler structure especially without making pattern of metallic structures. Numerical results show that the absorption bandwidth is about 1.469e-2 THz and the total quality factor *Q*, defined as Q = *f*_{0}/Δ*f*, reaches about 94.6, while the narrowband of the absorber can be attributed to the low power loss of the guided mode resonance in the dielectric layer. The simulation results are analyzed with coupled mode theory. Interestingly, by tuning the Fermi energy of BDSs from 50 to 80 meV, the absorption frequency can be tuned from 1.381 to 1.395 THz while the absorption peak is maintained at a level greater than 0.95. An additional numerical study elaborates on the dependence of the absorption on the diameter, height, period, surrounding medium with different refractive index and incidence angle.

## 2. Structure and materials

Figure 1 shows the schematic of the proposed structure consisting of photonic crystal slabs with a thick BDSs films. The periods of the photonic crystal slab in the *x* and *y* directions are *P _{x}* and

*P*, respectively. The height of the photonic crystal slab is

_{y}*H*and the circular air hole with a diameter of

*D*is introduced into each unit cell. In order to eliminate the transmission totally to achieve perfect absorption, the thick of the lower BDSs

*t*should be set larger than the skin depth of the electromagnetic waves and we set

*t*= 20 μm which is much enough to make sure it is opaque. And then, only the reflection of the absorber needs to be considered, the absorption can be obtained by

*A*= 1 -

*R*. Attribute to its 90° rotational symmetry, the proposed absorber have the attractive property of being polarization-independent at normal incidence. The absorber, unless otherwise indicated, is illuminated by

*x*-polarized (electric field E is along the

*x*axis) plane waves at normal incidence, as illustrated in Fig. 1. The refractive index (RI) of the photonic crystal slab is considered to be

*n*= 3.416. The complex conductivity of the BDSs was calculated by using the Kubo formalism in random-phase approximation theory (RPA) at the long-wavelength limit. Accounting to the intraband and interband contributed to the longitudinal, at the low-temperature limit

*T*<<

*E*, the dynamic conductivity of the Dirac semimetal can be written as [35]:

_{F}*g*= 40 is the degeneracy factor,

*k*=

_{F}*E*/

_{F}*ℏυ*is the Fermi momentum and

_{F}*υ*≈c/300

_{F}_{is the Fermi velocity, EF is the Fermi level, Ω = ℏω/EF + i ћτ−1/ EF, where τ = μEF/eυ2 F and μ is carrier mobility. εc = Ec/EF (Ec is the cutoff energy beyond which the Dirac spectrum is no longer linear). The real and imaginary parts of the dynamic conductivity of the Dirac semimetals which are normalized to 1nm of the thickness are shown in Figs. 2(a) and 2(b), respectively. As we can see from Fig. 2(a) that the real parts of the dynamic conductivities gradually decrease to almost zero with the increase of ℏω/EF, while they stepwise increase suddenly as ℏω/EF increased to 2, and then increase with the increase of ℏω/EF. As for the imaginary parts, they decrease with the increase of ℏω/EF and become negative as ℏω/EF increased to 1.23, while they increase with the increase of ℏω/EF when ℏω/EF is larger than 2. Fig. 1 Schematic of the proposed metasurface composed of the upper photonic crystal slab with circular air holes and the lower thick BDSs films. Corresponding electromagnetic excitation configuration (with polarization direction along the x axis) and geometric parameters are denoted by black letters in Fig. 1. Download Full Size | PPT Slide | PDF Fig. 2 (a) The real and (b) imaginary parts of the dynamic conductivity for BDSs (normalized to 1 nm of the thickness) at zero temperature in units e2/ℏ as a function of the normalized frequency ℏω/EF. The parameters of BDSs are set as g = 40, εc = 3, τ = 4.5 × 10−13 s. Download Full Size | PPT Slide | PDF Therefore, the conductivity of BDSs can be dynamically adjusted by tuning its Fermi energy EF which can be realized by alkaline surface doping in experiment [33, 34].Using the two-band model and taking into account the interband electronic transitions, the permittivity of the 3D Dirac semimetals can be expressed as [36]:(3)$$\epsilon ={\epsilon}_{b}+i\sigma /\omega {\epsilon}_{0}$$where εb = 1 for g = 40 (AlCuFe quasicrystals [37]) and ε0 is the permittivity of vacuum. Our results are obtained by using FDTD method, where periodic boundary conditions are applied along the x and y directions, and perfectly matched layers (PML) are considered in the z directions.3. Simulations and theoryTo illustrate its operating principle and performance, we numerically simulate the optical response of the proposed absorber from Fig. 1. The parameters are set as: Px = Py = 84 μm, H = 30 μm, D = 30 μm and EF = 65 meV. According to the solid red curve in Fig. 3(a), the simulated absorption spectrum demonstrates that the proposed absorber achieved perfect absorption at the resonance frequency of f0 = 1.3898 THz (λ0 = 215.71μm) and the FWHM (full width at half maximums) is Δf = 1.469e-2 THz. The total quality factor Q, defined as Q = f0/Δf, reaches about 94.6. To get the ultra-narrowband absorption mechanism, Figs. 3(b) and 3(c) demonstrate the simulated electric field |EZ|2 distributions of the proposed absorber at the frequency of 1.3898 THz. As shown, the maximum of the |EZ|2 is in the plane of the photonic crystal slab and just above the mirror, that’s to say, the electric field is mainly distributed in the lossless dielectric material. It means that the low power loss will occur in the resonance, thus resulting in an absorber possessing narrow bandwidth and high quality factor. In a word, the narrow bandwidth absorption of our absorber can be attributed to the low power loss in the guided mode resonance. To study the operating mechanism of the absorber theoretically, we describe the absorption spectrum of the absorber by utilizing coupled mode theory (CMT). According to CMT, the reflection coefficient of a resonator is [38–40]:(4)$$r=\frac{{s}_{-}}{{s}_{+}}=\frac{j\left(\omega -{\omega}_{0}\right)+\delta -{\gamma}_{e}}{j\left(\omega -{\omega}_{0}\right)+\delta +{\gamma}_{e}}$$and the absorption A = 1 - |r|2:(5)$$A=\text{}\frac{4\delta {\gamma}_{e}}{{\left(\omega -{\omega}_{0}\right)}^{2}+{\left(\delta +{\gamma}_{e}\right)}^{2}}$$where s+ and s- stands for the input and output waves of amplitudes, respectively, δ is intrinsic loss rate and γe is external leakage rate of the absorber. As shown as the above Eqs. (4) and (5), the physical properties of the resonator are fully determined by δ and γe. They reveal that perfect reflection, namely r = 1, is experienced when |ω - ω0| is much more than the sum of δ and γe. While when the resonator is on resonance, namely ω = ω0, and the external leakage γe is equal to intrinsic loss rate δ, no reflection is experienced and perfect absorption is achieved. The Eq. (4) is used to fit the absorption spectrum which is obtained by simulation, and they consist well with each other, as shown in Fig. 3(a). The fitting parameters are as follows: f0 = 1.3897 THz, γe = δ = 3.68e-3 THz. Then, the values of the absorptive and radiative quality factors, defined as Qδ = ω0/2δ and Qe = ω0/2γ, respectively, are approximate to 188.82. The total quality factor Q can also be obtained by CMT and be calculated from the equation: Q = Qe Qδ/(Qe + Qδ), and the value of Q in theory is 94.41. It is nearly the same as the Q obtained by numerical simulation, which prove that the realization of perfect absorption can attribute to the critical coupling. Fig. 3 (a) The absorption spectra of the proposed absorber. The red curve (blue point) is the numerical (theoretical) result achieved by the FDTD (coupled mode theory) method. (b) Electric field |EZ|2 distributions of the proposed absorber at the resonance frequency of f0 = 1.3898 THz. (c) Electric field |EZ|2 distributions of the cross section of the proposed absorber at the resonance frequency of f0 = 1.3898 THz. The air hole and the slab boundaries are indicated by the white lines. Download Full Size | PPT Slide | PDF To data, most of absorbers have been designed at a fixed frequency and the geometric parameters of the absorbers have to be carefully re-optimized when the application requirements of the change of absorption peaks. Next, we’ll investigate the frequency tunability of the proposed absorber. Simulations results in Fig. 4 show that when the EF of DBSs films is changed from 50 to 80 meV, the absorption frequency shifts from 1.381 to 1.395 THz while the absorption peak is maintained at a level greater than 0.95. To reveal the underlying physics mechanism of the tunability of the proposed absorber, the real and imaginary parts of permittivity of BDSs films for different EF as a function of frequency are plotted in Fig. 5. Fig. 4 Absorption spectra for different EF of the BDSs. Download Full Size | PPT Slide | PDF Fig. 5 (a) The real and (b) imaginary parts of the permittivity of BDSs. Download Full Size | PPT Slide | PDF As we can see from Fig. 5(a), the real parts of permittivity of BDSs slowly increase as the frequency increase from 1.35 to 1.45 THz when the EF keeps unchanged, while it decreases with the increase of the EF for the same frequency. Moreover, the imaginary parts of permittivity of BDSs slowly decrease as the frequency increase from 1.35 to 1.45 THz when the EF keeps unchanged and it increases with the increase of the EF for the same frequency, as shown in Fig. 5(a). According to perturbation theory, the change of resonance frequency of the absorber caused by material perturbation of the BDSs films can be estimated using the equation [41–43]:(6)$$\frac{\Delta \omega}{{\omega}_{0}}=\frac{\omega -{\omega}_{0}}{{\omega}_{0}}\approx \frac{-{\displaystyle {\iiint}_{V}dV\left[\left(\Delta \overrightarrow{\epsilon}\cdot \overrightarrow{E}\right)\cdot {\overrightarrow{E}}_{0}^{*}+\left(\Delta \overrightarrow{\mu}\cdot \overrightarrow{H}\right)\cdot {\overrightarrow{H}}_{0}^{*}\right]}}{{\displaystyle \iiint dV\left(\epsilon {\left|{\overrightarrow{E}}_{0}\right|}^{2}+\mu {\left|{\overrightarrow{H}}_{0}\right|}^{2}\right)}}$$where the numerator and denominator of the right hand side of the Eq. (6) represent the change of electromagnetic energy caused by the material perturbation and unperturbed total energy, respectively. Δ$\overrightarrow{\epsilon}$ and Δ$\overrightarrow{\mu}$ are the change in permittivity and permeability of the BDSs, respectively. ${\overrightarrow{E}}_{0}$ and $\overrightarrow{E}$ are the unperturbed and perturbed electric fields, respectively. ${\overrightarrow{H}}_{0}$and$\overrightarrow{H}$ are the unperturbed and perturbed magnetic fields, respectively. As the EF of BDSs increases, the real part of permittivity of BDSs decreases, namely Δε is less than 0 and thus results in the blue shift of the resonance frequency of the proposed absorber (Δω > 0).We also investigated the effects of the geometric parameters of the proposed absorber on the absorption performance. Figures 6(a)-6(d) show the absorption spectra of the proposed absorber with different diameter D, height H, period P, surrounding medium with different refractive index n, respectively, and the other geometric parameters are fixed as Fig. 1. As shown in Fig. 6(a), the resonance frequency is pushed down as diameter is decreased. Conversely, increasing height pushes the resonance to lower frequency, as shown in Fig. 6(b). The reason for the red shift of the absorber may due to the increase of the effective refractive indices of the photonic slab as diameter is decreased or height is increased. The effects of the period of the proposed absorber on the absorption performance are also investigated. As period is increased, the resonance frequency is pushed down, as shown in Fig. 6(c). Similarly, increasing refractive index n of surrounding medium pushes the resonance to lower frequency, as shown in Fig. 6(d). It indicates that the proposed absorber have potential in sensing. Fig. 6 Absorption spectra for (a) different diameter D, (b) different height H, (c) different period P, and (d) surrounding medium with different refractive index n, respectively. Download Full Size | PPT Slide | PDF Finally, the absorption stabilities of the proposed absorber under oblique incidence angles for both TE and TM polarizations have been further investigated, as shown in Figs. 7(a) and 7(b), respectively. In Fig. 7(c) we show the absorption spectra of the proposed absorber under oblique incidence angles for TE polarizations. We note that the resonance frequency of the absorber exhibit minimal blue shift and the maximum absorption nearly no change when the incidence angle is varied smoothly from 0° to 30°. For the TM-polarized case, the resonance exhibit frequency splitting, and the two modes are labeled as R1 and R2, respectively, as shown in Fig. 7(d). As the incidence angle is increases from 0° to 30°, the resonance frequency of R1 mode slightly increase and the absorption peak almost reduce to 0, while the R2 mode exhibit red shift and the maximum absorbance is greater than 0.9 even the incidence angle is up to 30°. Therefore, we can conclude that the absorption performance is more unstable to angle variation under TM-polarization than that under TE polarizations. In order to interpret the R1 and R2 mode more clearly, we plot the magnetic field HZ distributions for the absorber illuminated by TM-polarized plane wave with the incidence angle θ = 15°. Figures 7(e) and 7(f) show the HZ distributions at the resonance frequency of R2 mode (fR2 ≈1.31 THz) and R1 mode (fR1 ≈1.40 THz), respectively. The HZ of R1 and R2 are similar with each other except that they are anti-phased, which confirm the frequency splitting mentioned above. Fig. 7 The schematic diagrams of oblique incidence plane wave for (a) TE polarizations and (b) TM polarizations, respectively. Absorption spectra for different incident angle with (c) TE polarizations and (d) TM polarizations, respectively. The magnetic field HZ distributions are shown at the resonance frequency of (e) R2 mode (fR2 ≈1.31 THz) and (f) R1 mode (fR1 ≈1.40 THz) for the absorber illuminated by TM-polarized plane wave with the incidence angle θ = 15°. The air hole boundaries are indicated by the white lines. Download Full Size | PPT Slide | PDF 4. Summary and conclusionIn this paper, a BDSs based tunable narrowband absorber at terahertz frequencies was proposed and it had the attractive property of being polarization-independent at normal incidence because of its 90° rotational symmetry. The proposed absorber not only combined new materials to achieve tunable frequency, but a simpler structure especially without making pattern of metallic structures, which was different from reported metamaterials absorbers. Thanks to the low power loss of the guided mode resonance in the dielectric layer, the absorption bandwidth was about 1.469e-2 THz and the total quality factor Q reached about 94.6. The simulation results were analyzed by coupled mode theory and it fitted well with this theory. Interestingly, by varying the Fermi energy of BDSs from 50 to 80 meV, the absorption frequency could be tuned from 1.381 to 1.395 THz while the absorbance was maintained at a level greater than 0.95. An additional numerical study elaborated on the dependence of the absorption on the diameter, height, period, surrounding medium with different refractive index and incidence angle. Our results may also provide potential applications in optical filter and bio-chemical sensing.FundingNational Natural Science Foundation of China (Grant Nos. 61505052, 61775055, 61176116, 11074069).References and links1. N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008). [CrossRef] [PubMed] 2. Y. Wang, B. S. Liu, R. B. Bian, W. Z. Mao, C. X. Liu, B. Ma, and L. 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