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Abstract
We investigate the coherent absorption properties of bulk Dirac semimetal (BDS) metasurfaces. The calculation results show that two coherent beams with the same intensity are irradiated from the opposite direction to the BDS metasurface, inducing the surface plasmon resonance in the terahertz band. By adjusting their relative phase, the absorption intensity can almost be tuned consecutively from 99.95% to less than 0.01%. In addition, the intensity and peak frequency of the coherent absorption can be dynamically adjusted through Fermi energy of the BDS metasurface, with the frequency shifting from 1.24 to 1.36 THz. This research shows that BDS has potential application prospects in the design of coherent detectors and modulators.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Metasurfaces are defined as a class of artificial planar material whose thickness is less than wavelength, and they present extraordinary physical properties that natural materials do not possess. Metasurfaces can realize flexible and effective regulation of electromagnetic wave, and have the unique ability to enhance optical absorption [1]. Importantly, in recent years, the bulk Dirac semimetal (BDS), known as “three-dimensional graphene”, has become the focus of attention because of its high carrier mobility (9×106 cm2V−1s−1), which is far superior to graphene (2×105 cm2V−1s−1) [2]. Compared to graphene, BDS has lower intrinsic loss due to higher mobility. Additionally, BDS is easier to be processed and more stable in physical properties [3,4]. Therefore, Su et al. stimulated the surface plasmon of BDS [5] and applied it to surface plasmon coupled waveguide in terahertz band [6]. Shen et al. demonstrated the electromagnetic induced reflection effect of BDS [7], which further opened up the optical application of BDS in terahertz band. In addition, optical absorption plays an important role in application devices such as photodetectors, modulators and so on. People have a great interest in achieving total optical absorption [8]. This perfect absorption has been achieved in diverse structures, such as graphene metamaterials [8] and metallic particle arrays [9,10]. However, most of the perfect absorbers have only one port, and a single beam of light resonates with free electrons on the surface of the material. When two beams of light are coherent, the condition of two ports is similar to the critical coupling effect. The transmitted and reflected light of two beams of light interfere with each other separately, so the incident light can be well absorbed. This phenomenon is called coherent perfect absorption (CPA) [11]. This method provides a new and flexible way to cope with the perfect absorption of nanostructures [12,13].
In previous work, people focused on the coherent perfect absorption of graphene films. Zhang et al. used etched graphene periodic arrays to obtain 99.93% coherent absorptivity [14]. Fan et al. combined with the angle sensitivity of graphene sheet to analyze CPA in terahertz band [15]. There are also a lot of researches on etched or complete graphene sheet [1,16,17], graphene ring periodic array [18] and graphene ribbons [19–21]. Graphene has a weak interaction with light due to its monolayer structure. Some complex nanostructures proposed by previous studies, such as nanoantenna array and cross-shaped resonator pair array, have made some progress in enhancing the effect of graphene and light [22–24]. Nevertheless, the designs of these structures is really intricate and the range of working frequency is finite. As a result, they are difficult to play a good role in practical applications. Metal-dielectric multilayer composites have also been proposed to achieve CPA, but with a larger thickness [25]. BDS, as a hot novel material in recent years, attracts us deeply so we take it into consideration. Because BDS not only is relatively easy to manufacture but also has lower intrinsic loss and more stable physical properties in terahertz band. In addition, compared with graphene, the frequency range of CPA is larger, approximately ranging from 1.2 THz to 1.4 THz. Moreover, the surface state of BDS is similar to that of graphene. It can actively change Fermi energy by electrostatic doping or gate voltage [26], and then dynamically adjusts the complex surface conductivity. Therefore, stable BDS has a good application in terahertz band like graphene in dynamically tunable surface plasmon optical devices [27–33]. However, up to now, few researches have reported on the dynamically tunable CPA effect based on BDS in terahertz band.
In this letter, we have explored the dynamically tunable CPA effect based on BDS metasurface in terahertz band. We present that the strong plasmon resonances in terahertz band are shown in periodical embedded medium of the BDS metasurface under two coherent incident beams. By controlling the relative phase of two coherent input beams, resonance absorption in the BDS will be enhanced or suppressed. Of course, besides CPA, coherent perfect transparency (CPT) can be achieved. In addition, because the Fermi energy of Dirac semimetal can be changed by electrostatic doping and applying gate voltage, the absorption intensity can realize dynamically tunable. The absorption peak frequency of CPA shift from 1.24 to 1.36 THz, and the absorptivity can be continuously changed from 99.95% to 0.01%.
2. Methods
Figure 1(a) shows the periodic BDS metasurfaces with a thickness of 0.2 µm, both sides of which are covered by air. In order to facilitate theoretical research, the medium is simplified and assumed to be air. The unit of BDS metasurface is shown in Fig. 1(b), where the width is s = 5 µm, the length is l = 80 µm, the periodic width in the y direction of the unit is Py = 210 µm, and the periodic length in the x direction is Px = 220 µm. On the basis of the random-phase approximation theory, the complex surface conductivity of BDS can be expressed by Kubo formula under the long-wave limit, which is a combination of in-band and inter-band processes. When the temperature T is non-zero, it can be obtained [34]:
Fig. 1. (a) schematic diagram of coherent Dirac Semimetal metasurfaces. Two coherent counter-propagating beams are illuminated perpendicularly from the opposite sides on the Dirac semimetal films. (b) a unit cell of the nanostructure. The geometric parameters of proposed structure are Px = 220 μm, Py = 210 μm, l = 80 μm, and s = 5 μm.
Our simulation is based on three-dimensional FDTD method (Lumerical FDTD Solutions). Two coherent parallel beams with equal intensity and frequency are set up, which are symmetric with respect to x- axis and in opposite directions. And the values of I+ and I− can be both set to any identical numbers. In the simulation, BDS imported by sampled data is modelled as a conductive metasurface with a thickness of 0.2 µm and the lattice constant is set to be Px = 220 µm and Py = 210 µm respectively. As far as boundary conditions are concerned, we consider periodical boundaries in the x and y directions and a perfectly matched layer (PML) are adopted in z-axis. The light source illuminates the proposed structure along the z direction with the polarization of the electric field along the x direction.
3. Results and discussion
Two counter-propagating and coherent modulated beams (I+ and I−) are incident vertically from both sides to the surface of the structure, with O+ and O− being the respective output elements. The complex scattering coefficients (O±) of the BDS metasurface can be linked to the two incident beams (I±) by a scattering matrix, S, which is defined as [1]:
In general, the transmission and reflection of perfect metamaterial absorbers disappear, so they can exhibit perfect absorption. Nevertheless, it is difficult to maintain high absorption on the surface of BDS. Fortunately, it is worth noting that the equation (7) provides a feasible method to obtain an additional condition for coherent perfect absorption with |t| = |r| and 1 + cosΔφ1cosΔφ22|I+I-|/(|I+|2+|I−|2) = 0, which is |t| = |r| and |I+| = |I−|. This demonstrates that the coherent modulation of the incident lights can restrain the scattering, thus exciting the complete absorption of coherent beams to realize CPA where (i) two input beams are of the same amplitude and (ii) the amplitudes of transmission and reflection are equal and the phase difference between them is mπ (m is an arbitrary odd number).
In addition, we calculated the simulated spectrum under the illumination of incoherent light. When only one side of the light is irradiated onto the BDS metasurface, one part of the energy is absorbed, the other part is reflected, and the remaining energy is transmitted through the structure, as shown in Fig. 2(a). We can clearly see that the surface plasmon resonance occurs at the frequency of 1.34 THz, and the maximum absorption intensity of energy reaches 49.9%, which is almost the theoretical maximum absorption coefficient. A beam of light can cause such a strong resonance, so CPA is likely to occur in the BDS metasurfaces we proposed. Simultaneously, we checked the phase of transmission and reflection coefficients as shown in Fig. 2(b). We can see that the phase difference between transmission and reflection is close to π at 1.34 THz, which meets one of the conditions of coherent absorption: |t| = |r|. Furthermore, when two coherent lights of the same intensity are simultaneously illuminated from both sides of the BDS metasurface, there is interference between two beams of light, resulting in coherent absorption.
Fig. 2. (a) The simulated reflection, transmission and absorption intensity of the Dirac semimetal films illuminated by a single beam (or two incident beams detuning). (b) The phases of transmission and reflection coefficients.
We investigate that two coherent lights with the same intensity are simultaneously irradiated on the surface of the structure when the phase differences between them are 0 and π respectively, and the absorption and reflection intensity coefficients are shown in Fig. 3(a). When the phase difference is 0, they are symmetrically distributed on both sides of the BDS metasurface, which is called even mode. The interference of even mode beams inhibits the scattering of light, which leads to the high localization of light on the surface of the structure and the increase of absorption intensity.
Fig. 3. (a) Normalized total scattering output intensity are illuminated by two coherent incident lights with the same intensity. Solid lines and dashed lines present even and odd modes respectively. (b) Phase difference between even reflection and odd reflection. (c) The figure of electric field at the perfect absorption peak f = 1.34 THz, the absorption valley f = 1.41 THz at (d), and the second absorption peak f = 1.63 THz at (e).
It is noteworthy that the reflection intensity is close to 0 and the absorption intensity can reach 99.97% at the frequency of 1.34 THz. At this resonance frequency, that is the even mode, and the phase difference is nearly zero, as shown in Fig. 3(b). In addition, at the frequency of 1.41 THz, the phase difference between two coherent beams is π, and we call it odd mode. Because of the enhancement of light scattering, it will reduce the absorption coefficient to the minimum of 0.01% at the resonance frequency, and the reflection coefficient will reach 50.0%. That is to say, we can consider that all the light passes through the BDS metasurface and forms a coherent transmission phenomenon. Therefore, we can see that the phase difference of the coherent light will affect the absorption intensity of the BDS metasurface by the simulated spectrum. Even mode will lead to CPA and odd mode will give rise to CPT.
In order to consider the variation of the energy distribution of electric field when coherent perfect absorption occurs on the BDS metasurface, Figs. 3(c)–3(e) present respectively the electric field intensity at f = 1.34 THz, f = 1.41 THz, and f = 1.63 THz. It can be clearly seen from Figs. 3(c) and 3(e) that the collective oscillation of free electrons on the surface of BDS and the incident light cause the dipole resonance at the resonance frequency, which realizes the enhancement of energy locality. Compared with Fig. 3(d) at non-resonance, the local energy in Fig. 3(c) increases by 500 times, thus realizing the CPA.
When the phase difference of two coherent beams irradiated on the BDS metasurface changes from 0 to 4π, the absorption coefficient at the corresponding resonance frequency f = 1.34 THz is fitted by our formula as shown in Fig. 4. As the phase difference varies from 0 to π, the absorption intensity changes from 99.95% to 0.01%. The coherent absorption intensity varies periodically with the phase difference, and the corresponding period is 2π. CPA and CPT can only occur in even mode and odd mode. When the phase difference is a non-integer multiple of π, the absorption coefficient is in an excessive state between 0 and 1, which satisfies the periodic change.
Fig. 4. Phase modulation of coherent absorption under two coherent beams of the same intensity to the Dirac semimetal films from opposite sides.
To find out more clearly that the influence of relative intensity of coherent light on the absorption coefficient, Fig. 5(a) shows the trend of absorption intensity at resonance frequency as the relative intensity of coherent light in even mode on both sides of the BDS metasurface changes. When I+/I− = 0 (irradiated by only unilateral light), the absorption coefficient reaches 49.9%, which corresponds to the simulation results of Fig. 2. With the increase of I+/I−, the absorption coefficient also increases. When I+/I− = 1, the absorption coefficient reaches 99.95%, which could be the coherent perfect absorption shown in Fig. 3(a). This simulation result agrees well with the theoretical results expressed by equation (7). And we can intuitively find that CPA can be achieved when light intensity is equal.
Fig. 5. (a) The absorption coefficient changes with the ratio of the intensity of two coherent beams when there is no relative phase difference. (b) The absorption intensity varies with the Fermi energy Ef in the Dirac semimetal films. With the increase of Ef, the absorption peak frequency shifts blue.
Otherwise, the Fermi energy on the surface of the BDS depends on the charge carrier density n. It makes the coherent perfect absorption effect of the BDS metasurface more practical. And the electronic density of the BDS metasurface can be easily changed by applying voltage and electrostatic doping. In this way, the coherent absorption intensity and frequency can be dynamically tunable without adjusting structural parameters and types of materials. Figure 5(b) shows the simulated spectra of the BDS metasurface with different Fermi energies. For Ef = 0.07 eV, the resonance frequency of surface plasmon is at f = 1.34 THz, which can realize CPA. When the doping concentration is changed slightly, the absorption intensity will change, as shown in Fig. 5(b). The absorption intensity first increases as Ef increases, reaches the maximum of 99.95% at Ef = 0.07 eV, and then decreases gradually. The larger the Fermi energy is, the smaller the absorption coefficient is. When the Ef is greater than 0.07 eV, we can draw a conclusion that the larger the Ef is, the smaller the absorption coefficient is. For instance, when Ef = 0.1 eV, the absorption coefficient is 73.10% and when Ef = 0.175 eV, the absorption coefficient is 39.27%. With the increase of Ef, the absorption peak frequency shifts blue. The plasmon resonance frequency can change from 1.24 to 1.36 THz. The phenomenon that blue shift of absorption peak increase with Ef is explained as follows. The conductivity of BDS is linearly related to Fermi energy. The increase of Ef means the increase of conductivity. Obviously, the conductivity properties of BDS will also be enhanced, which will make the electromagnetic field more concentrated and reduce the effective current path, and then lead to blue shift. Nevertheless, when Fermi energy exceeds 0.07 eV, the conductivity properties of BDS have reached a relatively high level, so its conductivity increases slowly with the increase of Ef. As a result, the blue shift has become less noticeable.
4. Conclusions
We have simulated and calculated the coherent absorption effect of the BDS metasurface. According to the change of relative phase of coherent beams on both sides, the coefficient of coherent absorption can be consecutively changed from 99.95% (CPA even mode) to 0.01% (CPT odd mode) at resonance frequency. As the phase difference changes periodically with a period of 2π, the absorption coefficient of the structure can be affected, where CPA can be realized when the light intensity is equal. In addition, the Fermi energy of BDS metasurfaces can be changed by electrostatic doping or applying voltage, so that the absorption intensity and resonance frequency can be dynamically adjusted. The simulation results can provide guidance for the coherent perfect absorption of BDS based metasurface in terahertz band in the future, which also has excellent application prospects in optical detection, optical modulation and signal processing and so on.
Funding
Hunan Provincial Innovation Foundation for Postgraduate (CX2018B226); National Natural Science Foundation of China (CX2018B226).
Acknowledgment
This work was supported by the National Natural Science Foundation of China (Grant Nos. 61505052 and 61775055) and Hunan Provincial Innovation Foundation for Postgraduate (CX2018B226).
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