1. Introduction

Topological insulator (TI) emerging as a novel promising type of quantum materials, has triggered intense research interest due to their unique dual energy states, namely insulating bulk state and metal surface state [1,2]. Such peculiar energy band structure makes TI exhibit a rich variety of distinctive physical properties, such as wide-band response, high mobility, and current-carrying massless Dirac electron, thus offering the unprecedented potential for new generation optoelectronic devices [3,4]. The optical absorption of TI is one of the most fundamental physical issues for the study of light-matter interactions, which directly influence the optoelectronic functionalities of TI-based devices [5–8]. It is of paramount significance to dynamically modulate the optical absorption of TI for its application in active optoelectronic devices, especially in photodetectors, modulators and switches [9,10]. Recently, highly confined Dirac plasmonic modes have been observed in TI, followed by the demonstration of the strategy to modulate the optical absorption of TI through localized surface plasmons [10,11]. However, those devices relied on such strategy suffer from the low absorption modulation amplitude (<25%) [12], since the plasmonic resonance can mainly be supported by the surface state of TI at infrared wavelengths.

Different kinds of routes have been generally proposed to realize the optical absorption modulation based on TI film [13–15]. For instance, the strategy for modulating TI absorption based on Tamm plasmon polaritons has been demonstrated, which has been achieved by designing the TI coated photonic crystal structure [13]. By altering the thicknesses of the photonic crystal layer in TI coated photonic crystal structure, the absorption of TI can be enhanced from 26% to 78%. In addition, another method of absorption modulation derived from the electromagnetically induced transparency (EIT)-like effect based on multilayer systems composed of TI film and dielectric Bragg mirror with graphene defect layer has also been investigated [14]. By tuning the gate voltage on graphene of device, the absorption of TI has been dynamically tuned from 35% to 85%. Whereas, how to further improve and control the TI absorption still remains a major challenge for the development of high-performance TI-based optoelectronic devices. Coherent perfect absorption (CPA), a new approach for efficiently controlling absorption by manipulating the relative phases of interference wave from two ports, has received burgeoning amount of interest [16–20]. The graphene-based [21–25], black phosphorus-based [26,27], MoS₂-based [28], and three-dimensional Dirac semimetal-based [29,30] CPA devices have been designed to analyze the properties of absorption modulation, presenting an excellent performance of dynamic and flexible operation. However, it is still unknown that whether the CPA approach is effective to realize the high absorption modulation of TI.

In this paper, we theoretically and numerically investigate the dynamic absorption modulation of topological insulator Bi_1.5Sb_0.5Te_1.8Se_1.2 (BSTS) with high modulation depth employing two coherent beams at oblique incidence. BSTS has been proposed as a promising topological insulator applying in next-generation optoelectronic devices owing to its pronounced surface electronic transport and highly insulating bulk interior [31]. Under the illumination of TE-polarized wave irradiation, the optical absorption of BSTS-based CPA device can be enhanced or suppressed by adjusting the phase difference of two coherent beams, leading to the coherent perfect absorption and coherent perfect transmission. Moreover, benefiting from the oblique incidence of coherent beams, additional control over the high absorption(>90%) can be achieved in the visible and near-infrared region by tailoring the incident angle. It can provide additional degrees of freedom in realizing tunable functionalities of light absorption. It is also found that the quasi-CPA wavelength for TE mode is capable of being effectively tailored via tuning the bulk thickness of BSTS film. By optimizing the bulk thickness, the maximum modulation depth can be achieved. The proposed TI-based CPA device is strongly anticipated to present an amazing application prospects in the field of signal detection, modulation and processing.

2. Methods

The schematic of the proposed BSTS-based coherent perfect absorption device is illustrated in Fig. 1(a), where two counter-propagating coherent beams are obliquely illuminated on the BSTS film from opposite sides. The topological insulator with the unique atomic structure can be regarded as an insulating bulk state coated with conducting surface states [32]. Figure 1(b) shows the atomic structure of BSTS, it can be seen that the compound consists of quintuple layers (QLs) blocks with alternating layers of Bi/Sb and Te/Se. The adjacent QLs are bounded together via weak van der Waals interactions, while the coupling realizes through strong covalent bond within a single QLs unit [33]. The relative permittivity of conducting surface layer can be represented by the Drude dispersion relation [34]:

 

Fig. 1. (a) Schematic of the proposed BSTS-based coherent perfect absorption device. (b) The atomic structure of BSTS. (c) Relative permittivity of conducting surface layer and insulating bulk layer for BSTS. Two coherent beams (I₁ and I₂) illumination on BSTS film for (d) TE mode and (e) TM mode.

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The insulating bulk layer is modeled by the Tauc-Lorentz dispersion formula whose dielectric function is given by:

Our simulation is based on Finite Difference Time Domain software (FDTD Solutions, Lumerical). In the simulation, two coherent beams with equal frequency are established, which are symmetric with respect to z-axis and in opposite directions. The incident light I₁ and I₂ obliquely illuminate the structure with the incident angle of θ, and the intensity values of I₁ and I₂ are set to any identical numbers. The thickness of conducting surface layer (t_sur) of BSTS film is assumed to be 1.5 nm, and the thickness of insulating bulk layer (t_bulk) of BSTS film is assumed to be 9 nm initially. In practice, the BSTS nanofilm may be epitaxially grown by catalyst-free physical vaper-phase deposition method. To determine the proper mesh size and ensure the convergence of simulation results, the convergence test was carried out by decreasing the mesh size until the simulation results don’t change anymore. The mesh size gradually increases outside the BSTS, where the maximum element size is set as 0.15 nm [35]. The medium on both sides of BSTS film is assumed to be air. Periodical boundary conditions are used in the y and z directions and a perfectly matched layer (PML) is adopted in x-axis.

3. Results and discussion

In the BSTS-based coherent perfect absorption device, the coherent incident light (I₁ and I₂) can be simply decomposed into TE and TM modes, as illustrated in Fig. 1(d)-(e). Under the TE-polarized wave illumination, electric field vectors are perpendicular to the incident plane and only contain y component, and magnetic field vectors contain z component. The electric field above and below the BSTS film can be described as [23]:

The magnetic field components above and below the BSTS film are given by:

For the symmetrical coherent beams of CPA system, we can get the relations k_a=k_b, $\sqrt {{\raise0.7ex\hbox{${{\varepsilon _\textrm{a}}}$} \!\mathord{\left/ {\vphantom {{{\varepsilon_\textrm{a}}} {{\mu_a}}}} \right.}\!\lower0.7ex\hbox{${{\mu _a}}$}}} = \sqrt {{\raise0.7ex\hbox{${{\varepsilon _\textrm{b}}}$} \!\mathord{\left/ {\vphantom {{{\varepsilon_\textrm{b}}} {{\mu_b}}}} \right.}\!\lower0.7ex\hbox{${{\mu _b}}$}}} = \sqrt {{\raise0.7ex\hbox{${{\varepsilon _\textrm{0}}}$} \!\mathord{\left/ {\vphantom {{{\varepsilon_\textrm{0}}} {{\mu_0}}}} \right.}\!\lower0.7ex\hbox{${{\mu _0}}$}}} = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {{Z_0}}}} \right.}\!\lower0.7ex\hbox{${{Z_0}}$}}$. Z₀=377Ω is the free-space impendence. Under oblique TE polarized wave incidence, the boundary conditions around the BSTS surface (x=0 plane) are ${E_{y,a}}|{_{x = 0}} - {E_{y,b}}|{_{x = 0}} = 0$, ${H_{\textrm{z},a}}|{_{x = 0}} - {H_{z,b}}|{_{x = 0}} = {\sigma _{TI}}{E_{y,a}}|{_{x = 0}} $ [23]. In the symmetrical structure of BSTS-based CPA device, the electric fields for incident light are assumed to be ${E_{\textrm{y},a,1}} = {E_0}$, ${E_{\textrm{y},b,2}} = {E_0}{e^{i\varphi }}$, where $\varphi$ is the relative phase of the coherent beams of I₁ and I₂. Considering the boundary conditions of electric and magnetic fields and derived from Eq. (4)–Eq. (7), the electric fields of two output beams can be represented by the equations:

The optical absorption intensity of BSTS film and incident wave intensity are written as:

Thus, the normalized absorption intensity of BSTS film can be described as:

Consequently, the absorption of BSTS-based CPA device is related to the relative phase of coherent beams, incident angle, and the complex conductivity of BSTS. Compared to the CPA device at normal incidence, the proposed BSTS-based CPA device can offer additional degrees of freed to modulate the light absorption of BSTS by changing the incident angle. The absorption reaches its maximum value when $\theta \textrm{ = arccos}({{\sigma_{TI}}{Z_0}/2} )$. Moreover, the complex conductivity of BSTS is the main factor for absorption due to its special energy band structure, resulting in the different absorption modulation properties compared to other materials.

While for TM-polarized incident wave, magnetic field vectors are perpendicular to the incident plane and only contain the y component, and electric field vectors contain the z component. The electric field components above and below the BSTS film can be described as:

Under oblique TM-polarized wave incidence, the bounding conditions around the BSTS surface are ${E_{\textrm{z},a}}|{_{x = 0}} - {E_{\textrm{z},b}}|{_{x = 0}} = 0$, ${H_{\textrm{y},a}}|{_{x = 0}} - {H_{y,b}}|{_{x = 0}} = {\sigma _{TI}}{E_{\textrm{z},a}}|{_{x = 0}} $. Accordingly, the optical absorption intensity and normalized absorption intensity of BSTS film can be expressed by the followed equations:

Here, the magnetic fields for incident light are assumed to be ${H_{y,a,1}} = {H_0}$, ${H_{\textrm{y},b,2}} = {H_0}{e^{i\varphi }}$, where $\varphi$ is the relative phase of the coherent beams of I₁ and I₂.

To investigate the characteristics of coherent perfect absorption at oblique incidence, we first theoretically calculated the absorption spectra of BSTS-based CPA device through Eq. (12) and Eq. (16), where the relative phase is assumed to be 2nπ (n=0,1,2…). The calculated absorption spectra at 1.31 μm with the varying angle of θ for TE and TM modes are illustrated as the blue lines in Figs. 2(a)–2(b), respectively. The calculated results illustrate that the BSTS-based CPA device exhibits different absorption properties for TE and TM mode at oblique incidence. The CPA can be achieved around 75° for TE mode at the quasi-CPA wavelength of λ=1.31 μm. However, the CPA doesn’t occur for TM mode at λ=1.31 μm. To prove the effectiveness of CPA in BSTS-based CPA device, we numerically simulated the optical response, the corresponding results are given as the red spheres in Figs. 2(a)–2(b). It is obvious that the simulation results agree well with the theoretical results. All of the subsequent results are derived from the numerical simulations. As depicted in Fig. 2(a), the absorption first gradually increases but then decreases as the incident angle increases for TE mode. The near perfect absorption of 99.998% can be achieved when the incident angle reaches 75°, corresponding to the CPA effect. Moreover, the high absorption (>90%) can be obtained within the incident angle ranging from 61° to 82° at 1.31 μm. In contrast, for TM mode, we can see from Fig. 2(b) that the maximum absorption can merely reaches 64.4% at 1.31 μm and keeps on decreasing to 0 with the incident angle increasing from 0° to 90°. For oblique incidence, the electric filed component of TM-polarized wave in the BSTS plane decreases as incident angle increases. Thus, a larger incident angle leads to a smaller in-plane electrical field, resulting in the lower absorption in BSTS-based CPA device.

 

Fig. 2. The absorption spectra of BSTS-based CPA device under different incident angle for (a) TE mode and (b) TM mode. The red spheres and blue solid line correspond to simulation and theoretical results, respectively. The absorption spectra as a function of incident angle and wavelength under the fixed phase difference of 2nπ (n=0,1,2…) for (c) TE mode and (d) TM mode.

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Next, the incident angle modulation characteristics for coherent optical absorption of BSTS-based CPA device were studied. The mapping diagrams of the absorption spectra with different wavelength for TE and TM modes are given in Figs. 2(c) and 2(d), respectively. For TE mode, by varying the incident angle from 0 to 80°, the device exhibits high absorption (>90%) within a broad range from 0.47 to 1.51 μm. For TM mode, the high absorption (>90%) can be realized within wavelength ranging from 0.47 to 0.94 μm. Accordingly, additional control over the high absorption can be achieved in the visible and near-infrared region by tailoring the incident angle. Moreover, the optical absorption of BSTS-based CPA device exhibits strongly dependent on the incident wavelength. It can be understood from Eqs. (12) and (16) that the absorption has a bearing on the complex conductivity of BSTS, where the value of complex conductivity changes gradually with the wavelength. The property of tunable high absorption in visible and near-infrared region could promote the promising application of BSTS-based CPA devices in the field of optical communication [38].

In order to further investigate the phase modulation characteristics of BSTS-based CPA device for TE mode, the mapping diagram of the absorption spectra as a function of phase difference and wavelength at the incident angle of θ=75° is depicted in Fig. 3(a). It can be seen that the absorption of BSTS-based CPA device can be actively and continuously tuned by the phase difference of coherent beams. Figure 3(b) shows the phase modulation spectra of coherent absorption at the quasi-CPA wavelength of λ=1.31 μm under the oblique incidence and normal incidence. At the incident angle of θ=75°, the absorption can be continuously modulated from 99.998% to 0.2% as the phase difference φ varies from 2nπ to (2n+1) π (n=0,1,2…). When φ=(2n+1)π, the absorption reaches nearly zero, corresponding to the coherent perfect transmission (CPT) effect. Such substantial reduction of absorption in CPT effect is attributed to the constructive interference of scattering fields escaping from the BSTS film. Under the normal incidence, the CPA can’t be achieved at λ=1.31 μm, but the absorption is still periodically modulated by the phase difference. This result indicates that the CPA is strongly dependent on the phase difference and incident angle of the two input coherent beams, as predicted by Eq. (12).

 

Fig. 3. (a) The absorption spectra as a function of phase difference and wavelength at the incident angle of 75° for TE mode. (b) Phase modulation of coherent absorption at the incident angles of 75° and 0°. The insets are corresponding to the destructive interference and constructive interference of two coherent beams. The electric field distribution |E_y|² in y direction at λ=1.31 μm for (c) CPT and (d) CPA.

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The CPA and CPT effect on BSTS-based CPA device can be furtherly explained by the standing wave theory. A standing wave generates when two coherent beams illuminating from opposite sides of BSTS film. The insets of Fig. 3(b) give the schematic diagram of destructive interference and constructive interference of two coherent beams. When BSTS film locates at the node of the standing wave, the destructive interference takes place and the interaction between electromagnetic field and BSTS film is weak. Therefore, the incident waves transmit through the film in a low loss manner with the occurrence of coherent perfect transmission. When BSTS film locates at the antinode of the standing wave, the interaction become stronger due to the constructive interference and the energy of incident waves can be absorbed completely, resulting in the coherent perfect absorption. We next observe the electric filed intensity distribution in y direction for CPT and CPA (the observation range in x direction is smaller than one period of standing wave), as depicted in Fig. 3(c) and 3(d), respectively. For CPT, the electric filed is mainly distributed in free space on both sides of BSTS film with the lowest electric field in the BSTS film. While for CPA, the electric field is highly confined in the space near BSTS film, and the maximum electric field intensity is achieved in BSTS film. Consequently, by simply varying the phase difference between two coherent beams to adjust the relative location of BSTS for node or antinode of standing wave, the BSTS film can be continuously switched from the high transparency state to strong absorption state.

The relative intensity of coherent beams(I₁/I₂) is another crucial factor for optical absorption modulation of CPA devices [39]. The absorption spectra as the function of relative intensity and wavelength under the fixed phase difference of 2nπ (n=0,1,2…) and incident angle of 75° is depicted in Fig. 4(a). It is found that with the fixed phase difference and incident angle, the absorption of BSTS-based CPA device can be simultaneously modulated by changing the relative intensity from 0 to 1. The extracted absorption spectrum with different relative intensity at quasi-CPA wavelength of λ=1.31 μm is displayed in Fig. 4(b). It is found that when I₁/I₂=0, corresponding to the single beam irradiation, the absorption has merely been achieved 49.997%. When the value of I₁/I₂ varies to 1, the absorption can be tuned to 99.998%, indicating that two coherent beams with similar intensity irradiation is the necessary formation condition of CPA in our proposed device. The reflection, transmission, and absorption spectra of BSTS-based CPA device under a single beam illumination has been presented in Fig. 4(c). It can be seen that when the same transmission and reflection are obtained (|r|²=|t|²), the absorption reaches the incoherent absorption limit with the maximum value of 49.997%, which corresponds to the results in Fig. 4(b). In addition, the absorption spectra under two beams irradiation and single beam irradiation have been depicted in Fig. 4(d) for comparison. It is obvious that BSTS-based CPA device exhibits the significant enhancement of absorption in the visible and near-infrared region after two coherent beams modulating. The reinforced interaction between the light and topological insulator is significant for the light-matter interaction investigation and low-energy optically controlling activities of TI-based optoelectronic devices.

 

Fig. 4. (a) The absorption spectra as the function of relative intensity and wavelength under the fixed phase difference of 2nπ (n=0,1,2…) and incident angle of 75°. (b) The absorption spectrum with the relative intensity varying from 0 to 1. (c) The reflection, transmission, and absorption spectra under a single beam illumination. (d) The absorption spectra under two beams irradiation and single beam irradiation.

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The coherent absorption properties of BSTS-based CPA device with different bulk thickness (t_bulk) of BSTS were finally investigated. Figures 5(a)–5(b) show the absorption spectra with different bulk thickness of BSTS under the fixed phase difference of 2π and incident angle of 75° for TE mode and TM mode, respectively. We can see that by adjusting the bulk thickness of BSTS film, CPA can be achieved with different quasi-CPA wavelength for TE mode, while the absorption increases with the bulk thickness increasing from 7 nm to 11 nm for TM mode. To be more intuitive, the extracted maximum absorption and the corresponding wavelength under different bulk thickness for TE mode and TM mode are depicted in Fig. 5(c)-5(d), respectively. For TE mode, with the bulk thickness of BSTS film ranging from 7 nm to 11 nm, the absorption keeps greater than 99.99%, along with a redshift of quasi-CPA wavelength from 1.16 nm to 1.47 nm. For TM mode, the maximum absorption can merely reach to 76% with the bulk thickness of 11 nm.In addition, the modulation depth has been employed to quantitatively evaluate the modulation capability of CPA devices, which can be defined as M(λ)= max (I_S)/ min (I_S), with I_S being the normalized total output intensity, represented by the square of the scattering amplitudes denoted by I_S = |r + t|² [26]. Figure 5 (e) and 5 (f) depict the corresponding modulation depth of BSTS-based CPA device with different bulk thickness for TE mode and TM mode, respectively. It can be seen that the modulation depths with the different bulk thickness for TE mode maintains a level of 10⁴, which are greater than that for TM mode. Moreover, the maximum modulation depth (4.99×10⁴) can be achieved with the bulk thickness of 9 nm for TE mode, which is strikingly higher than the reported BSTS-based absorption modulation devices [13,14]. Thus, the proposed BSTS-based CPA device exhibits excellent modulation capability with the wide absorption modulation range and high modulation depth at oblique incidence.

 

Fig. 5. The absorption spectra with different bulk thickness of BSTS under the fixed phase difference of 2π and incident angle of 75° for (a) TE mode and (b) TM mode. The maximum absorption and the corresponding wavelength for (c) TE mode and (d) TM mode. The corresponding modulation depth with different bulk thickness for (e) TE mode and (f) TM mode.

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4. Conclusion

We have theoretically and numerically investigated the CPA effect on Bi_1.5Sb_0.5Te_1.8Se_1.2 (BSTS) film. It is found that the CPA can be achieved under two counter-propagating coherent beams with the same intensity illumination. Based on the destructive interference and constructive interference of two incident beams, the absorption of BSTS can be consecutively modulated from 0.2% (corresponding to CPT effect) to 99.998% (corresponding to CPA effect) at quasi-CPA wavelength of 1.31 μm by adjusting the relative phase of two coherent beams. Under the illumination of TE-polarized wave irradiation, the absorption can be maintained greater than 90% within a broad range of wavelength from 0.47 to 1.51 μm by tuning the incident angle. In addition, the quasi-CPA wavelength can be flexibly selected via changing the bulk thickness of BSTS film, and the maximum modulation depth of 4.99×10⁴ can be achieved. Overall, the designed BSTS-based CPA device with impressive performance offers a promising way to achieve the effective light absorption modulation, which would find various promising applications in photonic and optoelectronic devices.