λ /20-Thick cavity for mimicking electromagnetically induced transparency at telecommunication wavelengths

Abstract Optical cavities play crucial roles in enhanced light–matter interaction, light control, and optical communications, but their dimensions are limited by the material property and operating wavelength. Ultrathin planar cavities are urgently in demand for large-area and integrated optical devices. However, extremely reducing the planar cavity dimension is a critical challenge, especially at telecommunication wavelengths. Herein, we demonstrate a type of ultrathin cavities based on large-area grown Bi2Te3 topological insulator (TI) nanofilms, which present distinct optical resonance in the near-infrared region. The result shows that the Bi2Te3 TI material presents ultrahigh refractive indices of >6 at telecommunication wavelengths. The cavity thickness can approach 1/20 of the resonance wavelength, superior to those of planar cavities based on conventional Si and Ge high refractive index materials. Moreover, we observed an analog of the electromagnetically induced transparency (EIT) effect at telecommunication wavelengths by depositing the cavity on a photonic crystal. The EIT-like behavior is derived from the destructive interference coupling between the nanocavity resonance and Tamm plasmons. The spectral response depends on the nanocavity thickness, whose adjustment enables the generation of obvious Fano resonance. The experiments agree well with the simulations. This work will open a new door for ultrathin cavities and applications of TI materials in light control and devices.


Introduction
2][3][4][5] The dimensions of optical cavities are generally limited by the material characteristics and operating wavelength.Metallic cavities enable light control and functionalities at visible and near-infrared ranges with the excitation of localized surface plasmon resonance on hundred-nanometer structures. 6,7Silicon (Si) dielectric particles with several hundred nanometers can work as nanoresonators for functional devices at optical wavelengths. 8,9The complex geometries of metallic and dielectric cavities strongly depend on the precision fabrication process, limiting the fabrication of large-area devices.Ultrathin planar cavities play important roles in the large-area and integrated functional devices. 10,11High refractive index (HRI) materials attract considerable attention for achieving ultrathin nanocavities with strong light confinement and optical interference at the nanoscale. 12,13As conventional HRI materials, Si and germanium (Ge) semiconductors with refractive indices of ∼3.47 and ∼4.27 at 1550-nm wavelength have advanced the rapid developments of optical signal processing and communications. 14,15Ge material enables the realization of optical cavities with several hundred nanometers for photodetection at the telecommunication band. 16The materials with higher refractive indices are in great demand for acquiring extremely thin cavities operating at telecommunication wavelengths.8][19][20] The state of matter with topological property was first reported in two-dimensional HgTe quantum wells. 17Subsequently, many three-dimensional (3D) materials with topological surface states were reported, such as Bi 2 Te 3 , Sb 2 Te 3 , Bi 2 Se 3 , and their compounds. 19,202][23] TI materials will offer a promising prospect for achieving ultrathin optical cavities and their applications in novel optical effects and functional devices.
Electromagnetically induced transparency (EIT) is a counterintuitive quantum interference effect occurring in atomic systems. 25The EIT effect acts as an essential role in modern optical physics due to its significant applications in optical data storage, optical switching, and enhanced light nonlinearities. 26ven so, the practical applications of atomic EIT are limited by the rigorous requirements of low temperature or stable gas lasing conditions. 272][33] The EIT-like effects in coupled optical cavities can be used to realize advanced ultracompact devices, for instance, sensitive sensors, 32 3D rulers, 33 and optical switching. 34Mimicking EIT in TI material-based cavities will not only enrich the formation mechanism of EIT-like effects but also advance the novel optical applications of TI materials, which, however, have not been demonstrated until now.
Herein, we reported a type of ultrathin planar cavities based on the large-area grown Bi 2 Te 3 TI nanofilms, which possess obvious optical resonance in the near-infrared region.The experimental results demonstrate that the Bi 2 Te 3 TI material presents ultrahigh refractive index at near-infrared wavelengths, contributing to the realization of optical cavities that are tens of nanometers in thickness.The TI material-based nanocavity thickness is ∼1∕20 of resonance wavelength (∼λ∕20), much thinner than those of cavities based on the conventional HRI Si and Ge materials.We also experimentally observed an analogy to atomic EIT at telecommunication wavelengths by depositing the nanocavity onto a one-dimensional (1D) photonic crystal.A transparency window distinctly appears in the original cavity-induced resonance absorption spectrum owing to the coupling between the nanocavity resonance and Tamm plasmons.The spectral profile relies on the cavity thickness and can evolve to an asymmetric Fano resonance line shape.The experimental results agree well with the simulations.This work will pave a new pathway for ultrathin optical cavities as well as the applications of TI materials in the mimicking of EIT and optoelectronic devices.

Results
Figure 1(a) shows the 3D diagram of the cavity with a TI material nanofilm coating on a metallic layer.To explore the optical properties, we fabricated the Bi 2 Te 3 TI nanocavity by using a magnetron sputtering (MS) technique (see Appendix).Figure 1(b) depicts an angle-tilted cross-sectional scanning electron microscopic (SEM) image of a nanocavity with the Bi 2 Te 3 and silver nanofilms deposited on a SiO 2 substrate.d and t denote the thicknesses of Bi 2 Te 3 and silver films, respectively.A 2-nm SiO 2 layer was deposited to isolate the silver and Bi 2 Te 3 films for avoiding the formation of the Schottky barrier and charge transport. 35The optical and top-view SEM images reveal the large area and good flatness properties of MS-grown Bi 2 Te 3 nanofilms, respectively (see Fig. S1 in the Supplementary Material).Compared with the pulsed laser deposition, the MS technique enables the large-area growth of relatively flat Bi 2 Te 3 films. 36The MS method also presents the better flatness and higher growth rate for fabricating Bi 2 Te 3 films when compared with the atomic layer deposition. 37igure 1(c) depicts the absorption spectrum of a 42-nm Bi 2 Te 3 TI film on a 30-nm silver layer (i.e., d ¼ 42 nm and t ¼ 30 nm).The Bi 2 Te 3 TI thickness can be clarified using the atomic force microscope (AFM) measurement (see Fig. S2 in the Supplementary Material).We measured the reflection, transmission, and absorption spectra of a TI nanocavity using a spectrophotometer (see Appendix).In Fig. S3 in the Supplementary Material, a distinct reflection dip appears at 1465-nm wavelength (i.e., λ ¼ 1465 nm) for the cavity with a thickness of ∼λ∕20, which is derived from the Fabry-Perot resonance in the asymmetric nanocavity.Here, the flat silver film with a large area can reflect a part of the incident light.The silver-reflected light will generate destructive interference with directly reflected light from the Bi 2 Te 3 TI layer around the resonance wavelength. 13Thus, the reflection light of the cavity can be effectively suppressed.The cavity thickness of ∼λ∕20 is ultrathin, according to the reported cavities. 10,38,39The optical loss of Bi 2 Te 3 TI gives rise to strong resonance absorption at a broad wavelength range covering 1100 to 1800 nm, as depicted in Fig. 1(c).Moreover, we proposed to integrate the TI nanocavity onto a 1D photonic crystal, as can be seen in Fig. 1(d).The Bi 2 Te 3 TI nanocavity was deposited on a photonic crystal with alternately stacked Ta 2 O 5 and SiO 2 layers.The Ta 2 O 5 and SiO 2 layers were fabricated using electron beam deposition.Figure 1(e) shows the angle-tilted cross-sectional SEM image of a fabricated TI nanocavity/photonic crystal structure.We can see that the photonic crystal has good uniformity with the periodic number N ¼ 16.The thicknesses of Ta 2 O 5 and SiO 2 layers are 182 and 260 nm, respectively (i.e., d t ¼ 182 nm and d s ¼ 260 nm). Figure 1(f) depicts the absorption spectrum of TI nanocavity/photonic crystal structure with d ¼ 42 nm and t ¼ 30 nm.A distinct narrow dip appears around 1550 nm wavelength in the absorption spectrum.Thus, the original broad absorption of TI nanocavity exhibits a transparency window at the telecommunication wavelength, which is analogous to the EIT effect in atomic systems. 28,29Meanwhile, we characterized the grown Bi 2 Te 3 nanofilm using a transmission electron microscope (TEM).Figure 2(a) shows the TEM image of a grown 42-nm Bi 2 Te 3 TI film transferred on the supporter of a copper microgrid (see Appendix and Note 1 in the Supplementary Material).The chemical composition of Bi 2 Te 3 nanofilm can be confirmed using the energy-dispersive X-ray (EDX) measurement, which reveals the atomic ratio of Bi∶Te ≈ 2∶3 (see Fig. S4 in the Supplementary Material).The good quality of Bi 2 Te 3 can also be verified by the Raman shift spectrum (see Fig. S5 in the Supplementary Material). 40Figure 2(b) depicts the selected area electron diffraction (SAED) pattern of Bi 2 Te 3 nanofilm, which corresponds to the rhombohedral Bi 2 Te 3 phase. 41It also denotes that the MS-grown Bi 2 Te 3 nanofilm presents the preferred growth orientation of (015).The highresolution TEM (HRTEM) image in Fig. 2(c) illustrates multiorientation atomic lattice arrangements in the Bi 2 Te 3 nanofilm.The HRTEM image exhibits distinct fringes with some lattice spacing containing 0.321 and 0.220 nm, corresponding to the (015) and (110) planes of Bi 2 Te 3 , respectively.The continuous rings of sharp diffraction pattern and atomic lattice arrangements reveal the polycrystalline nature of the Bi 2 Te 3 nanofilm.We measured the optical constant of Bi 2 Te 3 TI using a spectroscopic ellipsometer (see Appendix and Note 2 in the Supplementary Material).To differentiate the surface and bulk states, the ellipsometer-measured results can be fitted with the layer-onbulk model. 42,43][44] From Fig. 2(d), we can see that the fitted complex refractive indices agree well with the measurement.Figures 2(e) and 2(f) depict the complex refractive indices of surface and bulk for the Bi 2 Te 3 TI.For the surface, the extinction coefficient k is larger than the refractive index n at the wavelengths from 230 to 1930 nm, exhibiting metal-like characteristic.For the bulk, the refractive index can exceed 6.0 at the wavelengths of >1385 nm, which is much higher than the refractive indices of Si and Ge.The surface thickness was fitted as ∼2.29 nm, approaching the estimated 2.5 nm for 3D TI. 45 The bandgap energy is ∼0.21 eV, which is close to the reported bandgap (see Table S1 in the Supplementary Material). 46The HRI property of our Bi 2 Te 3 accords with the reported result of singlecrystal Bi 2 Te 3 film, even though there is a certain difference in the optical constants. 47The ultrahigh refractive indices contribute to the ultrathin TI nanocavities at near-infrared range, similar to the Ge nanocavities at visible range. 13To our knowledge, this is the first report of a planar TI cavity with tens of nanometers in thickness at telecommunication wavelengths.
Subsequently, we measured the absorption spectra of TI nanocavities with different Bi 2 Te 3 thicknesses.As shown in Fig. 3(a), the resonant absorption wavelength presents a redshift The inset depicts a three-level system of the EIT-like effect.ω c stands for the nanocavity resonant frequency.δ is the frequency detuning between the nanocavity resonance and Tamm plasmons in silver/photonic crystal structure.γ 1 and γ 2 are the decaying rates of nanocavity resonance and Tamm plasmons, respectively.κe iφ is the complex coupling coefficient between the nanocavity resonance and Tamm plasmons with a phase retardation φ.
in the near-infrared region as d increases from 30 to 52 nm.Simultaneously, the absorption peak possesses an obvious rise.To verify the experimental results, the absorption spectra were theoretically calculated using the transfer matrix method (see Fig. S6 and Note 3 in the Supplementary Material). 48In the calculations, we employed the measured complex refractive indices of the Bi 2 Te 3 TI surface and bulk in Figs.2(e) and 2(f) as well as the measured optical constants of silver film in Fig. S7 in the Supplementary Material.As depicted in Fig. 3(b), there exists a nearly linear relation between the absorption peak wavelength and Bi 2 Te 3 TI thickness.As mentioned above, the TI nanocavity thickness is ∼1∕20 of the resonance wavelength.Moreover, we simulated the spectral response and field distributions of the TI nanocavities using the finite-difference time-domain (FDTD) method (see Appendix).We can see that the experimental measurements agree well with the theoretical and simulation results.Figure 3(c) depicts the intensity distribution of the magnetic field in the TI nanocavity at the absorption peak wavelength.It shows that the magnetic field is confined and distinctly enhanced in the nanocavity.As shown in Fig. 2(d), there are large extinction coefficients for the Bi 2 Te 3 TI at near-infrared wavelengths, which means that the TI nanocavity possesses high optical loss.Thus, the Fabry-Perot resonance in the asymmetric cavity can give rise to relatively strong light absorption in the near-infrared region. 13The absorption ratio between the Bi 2 Te 3 TI and silver nanofilms is ∼13∶1, revealing that the light absorption mainly stems from the Bi 2 Te 3 TI nanofilm [see Fig. S8(a) in the Supplementary Material].From Fig. S8(b) in the Supplementary Material, we can see that the TI bulk dominates in the light absorption of Bi 2 Te 3 nanofilm, and the surface layers induce a slight redshift of the resonance absorption peak.Moreover, we deposited a 30-nm silver film on the photonic crystal and measured the spectral response (see Fig. S9 in the Supplementary Material).The result demonstrates that there exists a narrow reflection dip and an absorption peak at the wavelength of ∼1560 nm, which results from the formation of Tamm plasmons in the silver/photonic crystal multilayer.Both the Bi 2 Te 3 ∕silver and silver/photonic crystal structures present relatively strong light absorption at the wavelengths from 1545 to 1580 nm.Intuitively, the TI nanocavity/ photonic crystal structure would present stronger absorption at these wavelengths, but a distinct transparency window appears around 1550 nm in the absorption spectrum, as shown in Fig. 1(f).0][51] But, the appearance of absorption dips in the Bi 2 Te 3 ∕silver∕photonic crystal structure is fundamentally different from the spectral response (reflection dips) in the Tamm plasmon structures, even with a poly(vinyl alcohol) layer on the metal. 50,51As mentioned above, this counterintuitive phenomenon can be regarded as an analogy to the EIT effect in atomic systems. 31,32,52It can be analyzed with a prototype three-level model, as depicted in the inset of Fig. 1(f).In the model, j1 > and j2 > stand for the upper states, and j0 > denotes the ground state.The light impinges on the Bi 2 Te 3 nanofilm and excites the resonance in the TI nanocavity, which is analogous to the transition from j0 > to j1 >. 31,53 Tamm plasmons can be generated by the evanescent field of nanocavity resonance, which will couple with each other.The coupling between the nanocavity resonance and Tamm plasmons is analogous to the transition between the states j1 > and j2 >.The two possible transition processes j0 >→ j1 >→ j2 >→ j1 > and j0 >→ j1 > will form the destructive interference.The interference coupling contributes to the appearance of a narrow transparency window in the broad absorption peak.Therefore, the EIT-like effect is generated due to the coupling between the Fabry-Perot resonance in the planar TI cavity and Tamm plasmons in the silver/photonic crystal structure.This is different from the generation mechanism of EIT-like transmission in previous plasmonic structures, e.g., the dielectric waveguide with silver nanoparticles. 54In Fig. 1(f), we can see some small grooves in the absorption spectrum around the transparency window.The small grooves correspond to the transmission peaks of the multilayer structure, which originate from the constructive interference at the output interface, not the EIT-like effect.We measured the spectral response of TI nanocavity/ photonic crystal structures with different Bi 2 Te 3 thicknesses d when t ¼ 30 nm.From Fig. 3(d), we can see that the EIT-like spectral profile is dependent on d.The EIT-like spectrum distinctly appears as d ranges from 42 to 52 nm, which denotes that the EIT-like behavior is robust to the small fabrication deviation of Bi 2 Te 3 thickness.However, an obviously asymmetric Fano-type line shape can be generated when d ¼ 30 nm.The formation of Fano resonance can be derived from the coherent interference between a quasi-continuum state (broadly spectral nanocavity off-resonance) and a discrete excited state (narrowly spectral Tamm plasmons).The numerical simulations are in excellent agreement with the experimental results, as depicted in Fig. 3(e).Figure 3(f) shows the magnetic field distribution of the TI nanocavity/photonic crystal structure with a 42-nm Bi 2 Te 3 film at the absorption dip wavelength (i.e., 1550 nm).It is found that Tamm plasmons between the silver film and photonic crystal can be excited with a distinct field enhancement.It verifies the nanocavity coupling-induced excitation of Tamm plasmons.
Interestingly, the coupling-induced Tamm plasmons can effectively reduce the strong light absorption in traditional Tamm plasmon systems, which will be beneficial for reinforcing light-matter interactions.Finally, we studied the dependence of the EIT-like spectrum on the silver film thickness t in the TI nanocavity/photonic crystal structure, as shown in Fig. 4(a).The theoretical calculations illustrate that the induced transparency window becomes narrower and shallower with the increase of t.To verify it, we fabricated the TI nanocavity/photonic crystal structures with t ¼ 15, 30, 40, and 50 nm when d ¼ 44 nm and measured their absorption spectra.
As depicted in Fig. 4(b), the experimental results verify the theoretical prediction of spectral evolution.To clarify the mechanism of the EIT-like effect, we theoretically fitted the measured absorption spectra with the two-coupled-oscillator (TCO) model (see Appendix).The results show that the decaying rate γ 1 of oscillator 1 (TI nanocavity) slightly changes as the Bi 2 Te 3 TI thickness varies from 15 to 50 nm.γ 1 is ∼1.02 × 10 15 rad∕s when t ¼ 30 nm.The decaying rate γ 2 of oscillator 2 (silver/photonic crystal structure) slowly increases with t and can approach 0.85 × 10 13 rad∕s when t ¼ 30 nm.The coupling strength κ between oscillators 1 and 2 rapidly decreases as t increases.κ reaches 5.5 × 10 13 rad∕s when t ¼ 30 nm, which is much less than the threshold coupling strength κ T [κ T ¼ ðγ 1 − γ 2 Þ∕4 ¼ 5.06 × 10 14 rad∕s].Therefore, the coupling between the nanocavity resonance and Tamm plasmons is located in the weak-driving regime when t ¼ 30 nm. 29 The coupling phase retardation φ approaches π in the TI nanocavity/photonic crystal structures with different t (see Fig. S10 in the Supplementary Material), verifying the destructive interference between the nanocavity resonance and Tamm plasmons.Because the coupling strength satisfies the criterion κ < κ T , the observed transparency windows in our systems are attributed to the EIT-like effect with the existence of large γ 1 and small γ 2 even when t declines to 15 nm, as depicted in Fig. 4(c).Thus, the EIT-like response can be achieved in the TI nanocavity/photonic crystal structure with a Bi 2 Te 3 film thickness of ∼λ∕25.The absorption dip can approach a low value of 6% at the 1550-nm wavelength when t ¼ 15 nm (see Fig. S11 in the Supplementary Material).It means that the EIT-like effect can effectively bypass the optical loss in the structure with a small silver thickness.

Discussion
The Bi 2 Te 3 TI possesses an ultrahigh refractive index at nearinfrared wavelengths, much exceeding the conventional HRI Si and Ge semiconductors [see Fig. 2(d)].The ultrahigh refractive index of Bi 2 Te 3 TI enables the realization of planar cavities with tens of nanometers in thickness, which can find promising applications in light-matter interaction, for instance, infrared optoelectronic conversion.From Fig. 1(c) and Fig. S3 in the Supplementary Material, we can see that the TI cavity with a 42-nm Bi 2 Te 3 film presents an ultrabroad resonance absorption of light with high efficiency of >50% at the wavelengths from 970 to 1850 nm.The nanocavities can effectively break the absorption limit of 50% for ultrathin free-standing film at nearinfrared range. 55As shown in Fig. S8 in the Supplementary Material, the light absorption mainly stems from the Bi 2 Te 3 TI nanofilm with a narrow-bandgap semiconducting property.The absorption ratio between the TI and silver nanofilms is ∼13∶1 around the resonance wavelength.The Bi 2 Te 3 TI cavities will contribute to the high-efficiency, broad-spectrum, ultrathin, and large-area near-infrared optoelectronic devices, e.g., telecommunication band photodetectors.
From Fig. 3(b), we can see that the resonance wavelength presents a linear redshift with the increasing Bi 2 Te 3 TI thickness.The resonance wavelength is ∼20 times as large as the TI nanocavity thickness (∼0.05λ), which is more remarkable than the subwavelength-thin (0.13λ) layer of Ge nanostructures. 55This property also enables the precision monitoring of nanofilm thickness in the deposition process. 13To achieve this kind of optical cavities, the thicknesses of Bi 2 Te 3 TI are much thinner than those of HRI Si and Ge semiconductors (see Fig. S12 in the Supplementary Material).For example, the Bi 2 Te 3 TI thickness is ∼53% (64%) of Si (Ge) thickness for optical resonance at the 1550-nm wavelength.Thus, the thickness of the Bi 2 Te 3 TI cavity can be improved by 34% (24.5%) compared with that of the Si (Ge) cavity.Moreover, the nanocavities may find applications in ultrathin structures for wavefront shaping, similar to Si metasurfaces. 8he TI nanocavities will offer new building blocks for realizing novel optical effects.Herein, we have innovatively integrated the TI nanocavity onto a photonic crystal with alternately stacked Ta 2 O 5 and SiO 2 layers.Interestingly, an analogy to the atomic EIT effect has been observed at telecommunication wavelengths.This EIT-like behavior is attributed to the destructive interference coupling between the resonance in TI nanocavity and Tamm plasmons in silver/photonic crystal structure.Compared with the conventional materials, the Bi 2 Te 3 TI with relatively high refractive index and extinction coefficient has the specific advantage for the generation of the EIT-like effect in this system (see Fig. S13 in the Supplementary Material).Besides the nanocavity thicknesses, the EIT-like response is also dependent on the incident light angle.From Fig. S14 in the Supplementary Material, we can see that the induced transparency wavelength exhibits a blueshift with the increase of incident light angle.This stems from the identical shift of the Tamm plasmon wavelength by altering the incident light angle.The shifted wavelength of s-polarized light is different from that of p-polarized light, which stems from the energy splitting between the s-and p-polarized Tamm states. 49The adjustment of the incident light angle offers an effective way for dynamic tunability of the induced transparency window.The induced transparency spectrum can also be tuned by controlling the polarization of light at oblique incidence, as shown in Fig. S14(f) in the Supplementary Material.This kind of cavity structures based on planar multilayers can effectively avoid the fabrication process and area limitations of the structures based on dielectric or metallic nanoparticles for the generation of EIT-like effects.The EIT-like effect presents good robustness to the alteration of the environmental refractive index (see Fig. S15 in the Supplementary Material).The TI nanocavity also enables the formation of EIT-like behavior by integrating with the thin Ge and silver layers (see Fig. S16 in the Supplementary Material).The thickness can be effectively improved when compared with the Ge-based cavity.This work will open a new door for ultrathin optical cavities and promote the applications of TIs in light control and optical functional devices.

Fabrication
The Bi 2 Te 3 TI and silver nanofilms were prepared by the MS deposition technique.The Bi 2 Te 3 film was deposited using radio-frequency (RF) sputtering with a pressure of 0.9 Pa and power of 70 W.The silver film was deposited using the direct current sputtering with a pressure of 0.6 Pa and power of 50 W.The feasible grown area can be set as 4 cm × 4 cm for the flat nanofilms in our MS deposition.The 2-nm SiO 2 layer was deposited by employing the RF sputtering with a pressure of 1 Pa and power of 100 W. The flowing Ar (100 sccm) was employed as the sputtering gas.The 1D photonic crystal with alternately stacked Ta 2 O 5 and SiO 2 layers was prepared on the polished SiO 2 substrate (Beijing Dream Material Technology Co., Ltd.) through electron beam deposition.

Characterizations and Measurements
The surface flatness of grown Bi 2 Te 3 TI nanofilm was characterized using the commercial SEM equipment (FEI Verios G4) and AFM system (Bruker Dimension FastScan).The AFM system was also used to test the thicknesses of deposited films.The material morphology of Bi 2 Te 3 TI nanofilm was characterized through the SAED pattern and HRTEM image, which were measured using the TEM equipment (FEI Talos F200X) with a voltage of 200 kV.The Raman spectrum of Bi 2 Te 3 TI was measured using a confocal microspectrometer (WITec Alpha300R) with a 532-nm laser.The complex refractive index of Bi 2 Te 3 TI in the optical region was tested by employing a spectroscopic ellipsometer (HORIBA UVISEL Plus) with a 70-deg incident light angle.The absorption, reflection, and transmission spectra of fabricated samples were measured using a spectrophotometer (Hitachi UH5700).

Simulations
To verify the experimental results, we employed the FDTD method to numerically simulate the optical spectra and field distributions of TI nanocavity and TI nanocavity/photonic crystal structures. 56In FDTD simulations, the bottom and top of structures were set as the perfectly matched layer absorbing boundary conditions.The other sides of computational space were set as the periodic boundary conditions.The non-uniform mesh was utilized in the y-axis direction to fully consider the influence of each layer.The mesh size of the Bi 2 Te 3 TI surface layer was set as 0.3 nm.The maximum mesh size of other layers was 5 nm.The absorption spectra of the entire structures were achieved with A ¼ 1 − jP r ∕P i j − jP t ∕P i j, where P r , P t , and P i denote the powers of reflected, transmitted, and incident light, respectively.The complex refractive indices of Bi 2 Te 3 TI surface and bulk were set as the measured values in Figs.2(e) and 2(f).The measured complex refractive index of MS-deposited silver film was used in the simulations (see Fig. S7 in the Supplementary Material).The refractive indices of SiO 2 and Ta 2 O 5 were set as 1.44 and 2.058 at the wavelengths of interest, respectively. 57,58The refractive indices of Si and Ge in the simulations come from the experimental data. 59,60

Theory
To clarify the mechanism of the EIT-like effect in the TI nanocavity/photonic crystal structures, we theoretically studied the EIT-like spectra with the TCO model. 31,53As shown in Fig. 1(c), the resonant mode in the TI nanocavity (i.e., oscillator 1) can be generated when the light is incident on the Bi 2 Te 3 TI nanofilm.Tamm plasmons in silver/photonic crystal structure (i.e., oscillator 2) can be excited by the resonant coupling of TI nanocavity.As depicted in the inset of Fig. 1(f), ω c stands for the resonant frequency of oscillator 1. δ is the frequency detuning between oscillators 1 and 2. γ 1 and γ 2 are the decaying rates of oscillators 1 and 2, respectively.κe iφ is the complex coupling coefficient between the two oscillators with a coupling phase retardation φ.When jω − ω c j ≪ ω c , γ 2 ≪ γ 1 ≪ ω c , and jδj ≪ γ 1 , the light absorption in the entire structure can be described as where AðωÞ is the imaginary part of the solution for the coupled differential equations of the two oscillators. 31,53f is an amplitude coefficient.According to Eq. ( 1), we can fit the experimentally measured absorption spectra to extract the related physical parameters.As shown in Fig. 4(b), the fitting curves are well consistent with the experimental results, verifying the feasibility of the theoretical model.

Fig. 1
Fig. 1 TI structures and absorption spectra.(a) 3D diagram of the ultrathin nanocavity with a TI nanofilm coated on a metallic layer.(b) Angle-tilted cross-sectional SEM image of a Bi 2 Te 3 TI nanocavity deposited on a SiO 2 substrate.The scale bar is 200 nm.Here, d and t are the thicknesses of Bi 2 Te 3 and silver nanofilms, respectively.(c) Experimental measurement of the absorption spectrum of the nanocavity with a 42-nm Bi 2 Te 3 film on a 30-nm silver layer (i.e., d ¼ 42 nm and t ¼ 30 nm).(d) 3D diagram of a TI nanocavity on a 1D photonic crystal.(e) Cross-sectional SEM image of a Bi 2 Te 3 nanocavity on a photonic crystal with alternately stacked Ta 2 O 5 and SiO 2 layers.Here, d t and d s are the thicknesses of Ta 2 O 5 and SiO 2 layers, respectively.N is the periodic number.The scale bar is 2 μm.The inset shows the high-resolution SEM image of Bi 2 Te 3 and silver films on the photonic crystal.The scale bar is 200 nm.(f) Measured absorption spectrum of the TI nanocavity/photonic crystal structure with d t ¼ 182 nm, d s ¼ 260 nm, d ¼ 42 nm, t ¼ 30 nm, and N ¼ 16.The inset depicts a three-level system of the EIT-like effect.ω c stands for the nanocavity resonant frequency.δ is the frequency detuning between the nanocavity resonance and Tamm plasmons in silver/photonic crystal structure.γ 1 and γ 2 are the decaying rates of nanocavity resonance and Tamm plasmons, respectively.κe iφ is the complex coupling coefficient between the nanocavity resonance and Tamm plasmons with a phase retardation φ.

Fig. 2
Fig. 2 Material characterization and optical constants of Bi 2 Te 3 TI.(a) TEM image of a grown 42-nm Bi 2 Te 3 TI film transferred on the supporter of a copper microgrid.(b) SAED pattern of the Bi 2 Te 3 nanofilm.(c) HRTEM image of the Bi 2 Te 3 nanofilm.(d) Ellipsometer-measured (circles) and fitted (curves) refractive indices and extinction coefficients of Bi 2 Te 3 TI at the wavelengths from 230 to 1930 nm.(e), (f) Fitted refractive indices and extinction coefficients of Bi 2 Te 3 TI surface and bulk states with the layer-on-bulk model [the inset in panel (d)].

Fig. 3
Fig. 3 TI thickness-dependent nanocavity resonance and induced transparency.(a) Experimentally measured absorption spectra of the TI nanocavities on the SiO 2 substrate with different Bi 2 Te 3 film thicknesses d when t ¼ 30 nm.(b) Corresponding resonance wavelengths of the TI nanocavity with different d .The curve, red circles, and blue circles denote the theoretical, experimental, and simulation results, respectively.(c) Corresponding magnetic field distribution of the TI nanocavity with a 42-nm Bi 2 Te 3 film (i.e., d ¼ 42 nm) at the absorption peak wavelength.(d), (e) Measured and simulated absorption spectra of the TI nanocavity/photonic crystal structures with different d when t ¼ 30 nm. (f) Corresponding magnetic field distribution of the TI nanocavity/photonic crystal structure with a 42-nm Bi 2 Te 3 film at the absorption dip wavelength.

Fig. 4
Fig. 4 EIT-like response with different silver film thicknesses.(a) Theoretical evolution of absorption spectrum from the TI nanocavity/photonic crystal structures with different silver film thicknesses t when d ¼ 44 nm.(b) Measured (solid curves) and fitted (dashed curves) absorption spectra of the TI nanocavity/photonic crystal structures with different t when d ¼ 44 nm.Here, the experimentally measured spectra were fitted with the TCO model.(c) Fitted decaying rates γ 2 and coupling strengths κ in the TCO model with different t.The inset shows the corresponding decaying rates γ 1 .