Topological insulators as relatively new quantum materials with the topologically protected conducting Dirac surface state reveal fantastic electronic and photonic characteristics. The photonic behaviors of topological insulators are particularly significant for exploring their optical phenomena and functional devices. Here, we present the generation of Tamm plasmon polaritons (TPPs) in a topological insulator multilayer structure consisting of a Bi1.5Sb0.5Te1.8Se1.2 (BSTS) nanofilm and a one-dimensional photonic crystal (PC). The results illustrate that the TPP electric field can locally concentrate between the BSTS nanofilm and PC, contributing to the improved light-BSTS interaction with a 3-fold enhancement of light absorption. It is also found that the near-infrared TPP response can be dynamically tailored by adjusting the PC layer thickness, BSTS nanofilm thickness, and angle of incident light. The theoretical calculations are in excellent agreement with the numerical simulations. Additionally, the TPP field intensity and light-topological insulator interaction are capable of being further reinforced by introducing a dielectric spacer between the BSTS nanofilm and PC. Our results will enrich the optical characteristics and application potential of topological insulators.

Topological insulators are a kind of newly emerging quantum materials, which attract growing attention in various fields, especially electronics, photonics, and optoelectronics. Because of strong spin-orbit coupling in the insulating bulk state, the topological insulators possess the topologically protected conducting surface (or edge) state, which is particularly different from ordinary insulators.1–11 The topological edge properties were first predicted and verified in the two-dimensional (2D) HgTe quantum wells.1,2 Subsequently, the topological surface states were found in the three-dimensional (3D) materials, such as Bi2Se3, Bi2Te3, and Sb2Te3.3–5 The one-dimensional (1D) and 2D Dirac cones can be formed in the helical edge and surface states of 2D and 3D topological insulators, respectively.6 The time-reversal symmetry of the Dirac surface (or edge) state can prevent the carriers from the back scatting when the currents encounter nonmagnetic impurities.1 Some excellent electronic behaviors have been reported in topological insulator materials, containing the quantum spin Hall effect, Majorana fermions, and topological magneto-electric effects.1,7 Topological insulators with the unique surface and bulk properties provide a good platform for quantum information processing, spintronics, and magneto-electric elements.1,7–9 Recently, a large amount of photonic response has been numerically and experimentally demonstrated in 3D topological insulators, such as saturable absorbing,12 nonlinear optical effects,13 photothermal conversion,14 photonic Weyl points,15 and nanometric hologram.16 Especially, surface plasmon polaritons (SPPs) in the ultraviolet, visible, and infrared regions were observed in topological insulators such as Bi2Se3, Bi2Te3, Sb2Te3, Bi1.5Sb0.5Te1.8Se1.2 (BSTS), and so on.17–28 Lupi et al. reported the terahertz plasmonic and magneto-plasmon generation from the micro-ribbon arrays of topological insulators, Bi2Se3.17,18 By using a scanning near-field optical microscope, Bao et al. observed the plasmonic response in the mid-infrared region from the Bi2Se3 sheets.19 Loh et al. demonstrated the visible-range surface plasmon modes in Bi2Te3 nanoplates.20 Zheludev et al. reported the ultraviolet and visible plasmonic modes in BSTS nanostructures.21,22 SPPs in topological insulators inject new vitality into the realization of optical functionalities, such as high-performance optical modulation,23 enhanced energy harvesting,24,25 angular-momentum nanometrology,26 and refractive index monitoring.28 Different from metals, the topological insulators present surface metal-like properties, which can support the generation of SPPs in an ultrabroad range from ultraviolet to terahertz.17–28 SPPs in topological insulators can be actively tuned by the optical pulse and possess lower propagation loss than the metals (e.g., silver and gold) in the ultraviolet range.23,26 Similar to metal-based SPPs, however, the excitation of SPPs based on topological insulators still requires the external assistance of grating, prism, or fiber/metal tips to compensate the wavevector mismatch between the SPPs and incident light.17–37 Tamm plasmon polaritons (TPPs), an optical state formed between the metal film and one-dimensional (1D) photonic crystal (PC), can be directly generated by optical excitation due to the zero in-plane wavevector.38 Besides this, the excitation of TPPs is independent of the polarization of incident light.38 The feasibly excited plasmon polaritons attract broad attention for the crucial applications in light-harvesting, nonlinear optical enhancement, lasing, optical emission, sensing, and strong coupling.39–47 Lee et al. presented the enhanced nonlinear optical response in the TPP structure with a weakly nonlinear optical material.40 Symonds et al. reported a novel laser on the basis of confined TPPs in the metal-semiconductor configuration.41 Leo et al. obtained the coherent emission with the generation of TPP modes by embedding silver patterns into an organic micro-cavity.42 The strongly enhanced light absorption of monolayer MoS2 was realized by means of the TPP excitation.43 TPPs were also proposed to realize the tunable multi-channel graphene-based perfect absorber operating in the terahertz region.44 Exploring the TPP response in topological insulators is particularly meaningful for revealing the optical properties of topological insulators and broadening their applications in photonics and optoelectronics.

Here, we numerically and theoretically investigate the near-infrared optical response from the topological insulator BSTS nanofilm coated on a 1D dielectric PC. The results illustrate that there exists an obvious dip in the reflection spectrum owing to the generation of the TPP mode between the BSTS nanofilm and dielectric PC. The localized TPP mode contributes to the 3-fold enhancement of light absorption in the topological insulator BSTS nanofilm. Moreover, we find that the TPP response is capable of being effectively tailored by altering the thicknesses of the PC layer and BSTS nanofilm. The adjustment of incident angle facilitates the tunability and selection of topological insulator TPP modes for both TM and TE polarized light. The light field intensity can be further reinforced by inserting a dielectric spacer between the BSTS nanofilm and PC, contributing to the stronger light-matter interaction. The theoretical results agree extremely well with numerical simulations. Our results will pave a new avenue for exploring light-topological insulator interaction and optical applications of topological insulators.

As depicted in Fig. 1, the topological insulator multilayer structure is composed of a BSTS nanofilm and a 1D PC. The topological insulator material is chosen as BSTS, which possesses good surface electronic transport and high bulk interior.22 The 1D PC consists of the alternately stacked silicon nitride (Si3N4) and silicon oxide (SiO2) layers. The light impinges on the BSTS nanofilm with an incident angle of θ. The thicknesses of BSTS, Si3N4, and SiO2 layers are denoted by dt, da, and db, respectively. The refractive indices of Si3N4 and SiO2 materials can be set as 2.2 and 1.45 in the wavelengths of interest, respectively. The period number of 1D PC is set as N. In practice, the proposed topological insulator structure is not difficult to be fabricated. The stacked Si3N4 and SiO2 dielectric layers can be deposited by plasma-enhanced chemical vapor deposition.48 The BSTS layer can be transferred by mechanically exfoliating from the bulk BSTS single crystals, which can be grown using a modified Bridgeman method.24 The reflection spectral features of the multilayer structure can be measured using the microspectrophotometer and microspectrometer.34 As shown in Fig. 1(b), the topological insulator can be considered as an insulating bulk state coated with conducting surface states.21 The thickness of the surface state layer can be assumed as ds = 1.5 nm.22 The relative permittivities of the bulk state and surface state layers can be achieved by fitting the experimental data with the Tauc-Lorentz and Drude models, respectively.20,21 For the insulating bulk state, the relative permittivity (εb = εb′ + b″) is derived from the interband transition, which can be calculated through Kramers-Kronig equations for its band structures. According to the Tauc-Lorentz model, the imaginary relative permittivity (εb″) of the insulating bulk state for the BSTS material can be described as

εbE=AE0CEEg2C2E2+E2E022Θ(EEg)E,
(1)

where Θ(EEg) = 0 for E < Eg and 1 for E > Eg. A, C, E0, and Eg correspond to the absorption peak amplitude, broadening factor, peak in joint density of states, and bandgap energy, respectively. E is the energy of incident photons. The real relative permittivity (εb′) of the insulating bulk state can be calculated by Kramers-Kronig integration

εbE=εb+2πPEgξεbξξ2E2dξ,
(2)

where εb(∞) is the parameter corresponding to the relative permittivity at high frequency. P stands for the Cauchy principal part of the integral. This equation can be written as the simplified form in Ref. 49. The relative permittivity of the conducting surface state for the BSTS material can be obtained by fitting with the Drude model εs(ω) = εs′ + s″ = εωp2/[ω(ω + )], where ε, γ, and ωp are the relative permittivities at the infinite frequency, electron collision frequency, and bulk plasma frequency, respectively. ω is the angular frequency of incident light in vacuum. According to the experimental measurement, these parameters can be fitted as A = 65.9, C = 1.94, E0 = 1.94 eV, Eg = 0.25 eV, εb(∞) = 0, ε = 1.3, ωp = 7.5 eV, and γ = 0.05 eV.21 As shown in Fig. 1(c), the bulk state exhibits lossy insulating characteristic (i.e., εb″ > 0) in the near-infrared region. Figure 1(b) shows that the surface state exhibits negative relative permittivities at this frequency range, which is similar to noble metals. Here, the transfer-matrix method (TMM) is employed to achieve the spectral response in the topological insulator multilayer structures. The TMM is an effective approach for the theoretical calculations of light propagation characteristics in the multilayer photonic configurations.38,43,45 In the TMM, Pi and Mi are assumed as the matrixes characterizing the light propagation passing through the ith layer and boundary, which can be expressed as

Pi=exp(jφi)00exp(jφi),
(3)
Mi=1ti1riri1,
(4)

respectively, where φi = 2πnidi cos θi/λ is the factor describing the light propagation phase in the ith layer. ni and θi represent the refractive index and angle of light propagation in the ith layer, respectively. They are determined by Snell’s law: ni sin θi = ni−1 sin θi−1 (θ0 = θ). di stands for the thickness of the ith layer. λ is the wavelength of incident light in vacuum. The incident light is set as the TM polarization. Based on Fresnel equations, the reflection and transmission coefficients are described as ri = (ni−1 cos θini cos θi−1)/(ni−1 cos θi + ni cos θi−1) and ti = 2ni−1 cos θi−1/(ni−1 cos θi + ni cos θi−1), respectively. By the multiplication of Pi and Mi, the total matrix in the TMM can be described as Q = M1P1M2P2P2N+3M2N+4. The transmission, reflection, and absorption spectra of topological insulator multilayer structures can be theoretically calculated by T = |1/Q11|2, R = |Q21/Q11|2, and A = 1 − TR, respectively. To explore the light propagation properties in the multilayer structures, the geometrical parameters are set as dt = 58 nm, da = 100 nm, db = 230 nm, N = 20, and θ = 0°. By employing the TMM, we theoretically calculate the reflection spectrum of the topological insulator multilayer configuration. As depicted in Fig. 2(a), a distinct dip appears in the reflection spectrum at the wavelength of 1068 nm. In the metal-PC structures, the appearance of this reflection dip is generally accompanied with the generation of a localized optical mode, namely, TPP mode.38,43 The negative relative permittivity of the conducting surface state for the BSTS nanofilm provides an essential condition for the excitation of the TPP mode in the BSTS-PC structure.38 To verify this, we use the finite-difference time-domain (FDTD) method to numerically simulate the light propagation characteristics in the topological insulator structures.50 Here, the numerical simulations are performed using the commercial FDTD software packages. In the FDTD simulations, the perfectly matched layer absorbing boundary conditions and periodic boundary conditions are set on the left/right and upper/lower sides of computational space, respectively.43 For the convergence of simulation results, the nonuniform mesh is utilized in different photonic layers. The mesh sizes of the surface state and bulk state layers are set as 0.3 nm and 1 nm, respectively. The reflection, transmission, and absorption spectra of topological insulator multilayer structures can be achieved using R = |Pr/Pin|, T = |Pt/Pin|, and A = 1 − RT in the FDTD simulations. Here, Pr and Pt stand for the light powers reflected and transmitted from the BSTS-PC structures, respectively. Pin is the power of incident light. Figure 2(b) shows the reflection spectrum obtained by the FDTD simulations, which is in excellent agreement with the TMM theoretical results.

FIG. 1.

(a) Schematic diagram of the topological insulator multilayer structure consisting of a BSTS nanofilm coated on a 1D PC with alternately stacked Si3N4 and SiO2 layers. dt, da, and db denote the thicknesses of BSTS, Si3N4, and SiO2 layers, respectively. (b) Equivalent layer-on-bulk model of the BSTS crystal with ultrathin conducting surface state layers (ds = 1.5 nm). (c) Relative permittivity of the bulk state underneath the surface state for the BSTS nanofilm. (d) Relative permittivity of the surface state for the BSTS nanofilm.

FIG. 1.

(a) Schematic diagram of the topological insulator multilayer structure consisting of a BSTS nanofilm coated on a 1D PC with alternately stacked Si3N4 and SiO2 layers. dt, da, and db denote the thicknesses of BSTS, Si3N4, and SiO2 layers, respectively. (b) Equivalent layer-on-bulk model of the BSTS crystal with ultrathin conducting surface state layers (ds = 1.5 nm). (c) Relative permittivity of the bulk state underneath the surface state for the BSTS nanofilm. (d) Relative permittivity of the surface state for the BSTS nanofilm.

Close modal
FIG. 2.

(a) Theoretical and (b) numerical results of the reflection spectrum in the topological insulator multilayer structure achieved by the TMM calculations and FDTD simulations, respectively. Here, dt = 58 nm, da = 100 nm, db = 230 nm, N = 20, and θ = 0°.

FIG. 2.

(a) Theoretical and (b) numerical results of the reflection spectrum in the topological insulator multilayer structure achieved by the TMM calculations and FDTD simulations, respectively. Here, dt = 58 nm, da = 100 nm, db = 230 nm, N = 20, and θ = 0°.

Close modal

Figures 3(a) and 3(b) depict the intensity distribution and profile of the electric field in the BSTS-PC structure at the reflection dip (λ = 1068 nm), respectively. We can see that the electric field is mainly confined between the BSTS nanofilm and PC, confirming the generation of the TPP mode in the topological insulator multilayer structure.38 The novel response of the topological insulator offers fantastic prospects for the enhancement of light-matter interaction. Figure 3(c) shows the light absorption spectra of the BSTS nanofilm with and without the 1D PC. It is found that the free-standing BSTS nanofilm possesses the relatively weak light absorption in the near-infrared region. The BSTS light absorption is stronger in the visible region, which may result from the bulk interband transitions.27 In the multilayer structure, the light absorption in the BSTS nanofilm can be particularly enhanced by the excitation of the TPP mode. As shown in Fig. 3(c), the BSTS light absorption can approach 78% at the TPP wavelength with a full width at half maximum (FWHM) of 129 nm. The TPP enhanced light absorption in the multilayer structure is 3 times larger than that of the free-standing BSTS nanofilm. The reinforced interaction between the light and topological insulator is significant for the low-energy optically controlling activities and optoelectronic devices based on topological insulators.23 To clarify the mechanism of light-BSTS interaction, we analyze the light absorption properties of the BSTS nanofilm by using the power dissipation density (PDD): w(x, y) = 0.5ε0ω[εs″(x, y)|Es(x, y)|2 + εb″(x, y)|Eb(x, y)|2].51 Here, Es(x, y) and Eb(x, y) stand for the electric field amplitudes in the surface state and bulk state layers of the BSTS nanofilm, respectively. They can be achieved by the FDTD simulations. The BSTS light absorption can be obtained by calculating the ratio of the absorbed power of the BSTS nanofilm (including absorbed powers from the surface and bulk layers) in the volume V to the power of light impinging on the BSTS surface area S,43,51

A=Vw(x,y)dV0.5cε0|Ein|2Scosθ,
(5)

where |Ein|2 is the electric field intensity of incident light. From Eq. (5), we can see that the BSTS light absorption is dependent on not only the electric field in the BSTS nanofilm but also the surface and bulk dissipative losses of the topological insulator. By using Eq. (5), we obtain the PDD analytical results of the BSTS light absorption. As shown in Fig. 3(c), the analytical results are consistent with theoretical calculations. The results in the inset of Fig. 3(c) illustrate that the light absorption of the BSTS nanofilm is mainly attributed to the lossy insulating bulk state in the near-infrared region, rather than the conducting surface state.

FIG. 3.

(a) Intensity distribution of the electric field in the topological insulator multilayer structure at the wavelength of 1068 nm. (b) Corresponding intensity profile of the electric field. (c) Absorption spectra of the BSTS nanofilm with and without the 1D PC. The circles are the PDD analytical results. The inset depicts the wavelength-dependent light absorption from the conducting surface state and insulating bulk state of the BSTS nanofilm.

FIG. 3.

(a) Intensity distribution of the electric field in the topological insulator multilayer structure at the wavelength of 1068 nm. (b) Corresponding intensity profile of the electric field. (c) Absorption spectra of the BSTS nanofilm with and without the 1D PC. The circles are the PDD analytical results. The inset depicts the wavelength-dependent light absorption from the conducting surface state and insulating bulk state of the BSTS nanofilm.

Close modal

Subsequently, we investigate the dependence of topological insulator TPPs on the thicknesses of the PC dielectric layer and BSTS nanofilm. Figure 4(a) depicts the evolution of the reflection spectrum with the thickness da of the Si3N4 layer in the near-infrared region. We can see that the TPP wavelength exhibits a linear red shift with the increase in da, which is similar to the response in metal-PC systems.43 Moreover, the spectral width of reflection dip increases with da. As shown in Fig. 4(b), the FDTD simulations agree well with the TMM theoretical results. When the thickness db of the SiO2 layer increases, the wavelength of the TPP mode also has a linear red shift (not shown here). Figure 4(c) depicts the evolution of the reflection spectrum with the thickness dt of the BSTS nanofilm. A slight red shift can also be observed for the reflection dip when dt increases from 40 nm to 75 nm. The TPP spectrum also exhibits a slow broadening with dt. The dependence of topological insulator TPPs on the geometrical parameters will contribute to the selective enhancement of light-matter interaction.

FIG. 4.

(a) Theoretical and (b) simulation results of reflection spectral evolution with the thickness da of the Si3N4 layer in the topological insulator multilayer structure with dt = 58 nm and db = 230 nm. (c) Theoretical and (d) simulation results of reflection spectral evolution with the thickness dt of the BSTS nanofilm when da = 100 nm and db = 230 nm.

FIG. 4.

(a) Theoretical and (b) simulation results of reflection spectral evolution with the thickness da of the Si3N4 layer in the topological insulator multilayer structure with dt = 58 nm and db = 230 nm. (c) Theoretical and (d) simulation results of reflection spectral evolution with the thickness dt of the BSTS nanofilm when da = 100 nm and db = 230 nm.

Close modal

The angle of incident light is a crucial factor for the dynamic tunability of TPP response.38 Here, we investigate the dependence of topological insulator based TPPs on the incident angles for TM and TE polarized light. As shown in Fig. 5, the TPP wavelengths exhibit a slight blue shift with increasing the incident angle θ. The FDTD simulations agree well with the TMM theoretical results. For the TE polarization, the reflection and transmission coefficients in Eq. (4) will change as ri = (ni−1 cos θi−1ni cos θi)/(ni−1 cos θi−1 + ni cos θi) and ti = 2ni−1 cos θi−1/(ni−1 cos θi−1 + ni cos θi), respectively. The TPP wavelength shift for the TE polarized light is different from that of the TM polarized light owing to the splitting between the TE and TM TPP modes.38,43 To explore further enhancement of light-matter interaction, we modify the multilayer structure by inserting an aluminum oxide (Al2O3) spacer between the BSTS nanofilm and PC. As depicted in Fig. 6(a), the spectral width of reflection dip decreases with the introduction of the spacer layer, while the reflection dip distinctly drops. The results in Fig. 6(b) illustrate that the reflection dip possesses a linear red shift with increasing thickness dc of the spacer layer. The theoretical results agree well with the numerical simulations. From Fig. 6(c), we can see that the light absorption of the BSTS nanofilm in the modified multilayer structure can be further reinforced to 85.3% with a FWHM of 89 nm. This absorption peak value in the modified structure possesses an improvement of 9.4% compared to the BSTS light absorption in the original multilayer structure. In Fig. 6(d), the electric field distribution at the TPP wavelength (λ = 1202 nm) illustrates that the light field mainly concentrates in the spacer layer, which is 1.85-fold stronger than that of the structure in Fig. 1(a). It reveals that the introduction of the spacer layer is meaningful for promoting the enhancement of light-matter interaction in the topological insulator TPP systems.

FIG. 5.

Evolution of the reflection spectrum with the incident angles for (a) TM and (b) TE polarized light in the multilayer structure with dt = 58 nm, da = 100 nm, and db = 230 nm. The circles denote the FDTD simulation results.

FIG. 5.

Evolution of the reflection spectrum with the incident angles for (a) TM and (b) TE polarized light in the multilayer structure with dt = 58 nm, da = 100 nm, and db = 230 nm. The circles denote the FDTD simulation results.

Close modal
FIG. 6.

(a) Reflection spectra of the multilayer structure with and without an Al2O3 spacer (i.e., dc = 0 nm and 400 nm) between the BSTS nanofilm and PC. (b) Evolution of the reflection spectrum with dc. The circles denote the FDTD simulation results. (c) Absorption spectrum of the BSTS nanofilm when dc = 400 nm. (d) Intensity distribution and profile of the electric field at the TPP wavelength (λ = 1202 nm) in the topological insulator structure with a spacer layer.

FIG. 6.

(a) Reflection spectra of the multilayer structure with and without an Al2O3 spacer (i.e., dc = 0 nm and 400 nm) between the BSTS nanofilm and PC. (b) Evolution of the reflection spectrum with dc. The circles denote the FDTD simulation results. (c) Absorption spectrum of the BSTS nanofilm when dc = 400 nm. (d) Intensity distribution and profile of the electric field at the TPP wavelength (λ = 1202 nm) in the topological insulator structure with a spacer layer.

Close modal

In summary, we have theoretically and numerically investigated the generation of TPPs in the topological insulator multilayer structure consisting of a BSTS nanofilm and a 1D PC. The TMM calculations and FDTD simulations illustrate that there exists a distinct dip in the reflection spectrum due to the excitation of the TPP mode between the BSTS nanofilm and PC, thereby giving rise to the 3-fold enhancement of light absorption for the BSTS nanofilm in the near-infrared region. It is found that the TPP wavelength and spectral width can be flexibly tailored by altering the geometrical parameters, containing the thicknesses of the PC dielectric layer and BSTS nanofilm. The adjustment of incident angles for both TM and TE polarized light also contributes to the dynamical tunability of TPP response. The theoretical results agree extremely well with the numerical simulations. Especially, we find that the light field intensity can be further improved by inserting a dielectric spacer between the BSTS nanofilm and PC, contributing to greater enhancement of light-matter interaction. The light absorption can be reinforced to 85.3% in the BSTS nanofilm. Our findings will pave a new avenue for enhancing light-matter interaction and enriching application potential of topological insulators in light harvesting and modulation.

This work was supported by the National Key R&D Program of China (Grant No. 2017YFA0303800), the National Natural Science Foundation of China (Grant Nos. 11774290, 11634010, and 61705186), the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2017JQ1023), and the Fundamental Research Funds for the Central Universities (Grant Nos. 3102018zy039 and 3102018zy050).

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