Plasmons of topological crystalline insulator SnTe with nanostructured patterns

Abstract

Using the finite-difference time-domain method and density functional theory, we theoretically investigate the plasmons of topological crystalline insulator (TCI) SnTe with nanostructured patterns. Due to the fact that the topological material has an insulating bulk surrounded by topologically protected metallic surfaces, the TCI nanogratings are uniquely described by the core–shell-like model. In contrast to the plasmons of metallic nanoparticles usually found in visible spectra, four plasmon resonances excited on the TCI SnTe nanogratings are numerically observed in the visible-near-infrared (vis-NIR) spectral region, which is in agreement with the theoretical analyses. Furthermore, periodic shifts of resonance wavelengths are achieved with the variation of grating heights. The facile control of the plasmons of TCI nanopatterns in the vis-NIR spectral region may have potential applications in biomedicine, plasmonic nanodevices, and integrated optoelectronic circuits.

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Introduction

Plasmons excited by massless Dirac fermions in graphene and topological insulators (TIs) have a pivotal role in the realization of integrated photonic devices and robust photo-detectors of radiation.^1–5 The massless Dirac fermions in graphene exhibit various plasmonic properties,⁶ such as in tunable terahertz metamaterials,⁷ persistent switching of graphene plasmons on a ferroelectric substrate,⁸ and even highly confined low-loss plasmons in graphene-boron nitride heterostructures.⁹ The exotic Dirac-type metallic surface states of TIs generated by strong spin–orbit coupling also attract great attention to plasmonic studies.⁴ Fano-like interference between the plasmons and the phonons is studied in a TI Bi₂Se₃ microribbon array.⁵ The blue–ultraviolet spectral range can be realized for plasmons in TI Bi_1.5Sb_0.5Te_1.8Se_1.2 with low loss, which is hardly obtained from noble metals.¹⁰ In general, Dirac plasmons greatly differ from the usual plasmonic research focusing on normal metallic materials,¹¹ e.g. gold and silver,^12–15 which may enable an array of exciting applications for plasmonics.

In contrast to TI surface states protected by time-reversal symmetry, topological crystalline insulators (TCIs), as a new state of quantum matter, possess surface states protected by crystalline (mirror) symmetry.^16–20 TCI materials were theoretically proposed in the SnTe-class of compounds.¹⁷ Then, they were experimentally prepared in Pb_1−xSn_xSe,^21,22 Pb_1−xSn_xTe,^23,24 and SnTe.²⁵ The surface states spanning the insulating bulk gap in TCIs have shown many exotic electronic properties,^17,26–28 such as the coexistence of massive and massless Dirac fermions,²⁶ and no back scattering even with nonmagnetic impurities.¹⁷ More importantly, various nanopatterns in TIs and TCIs have been realized in experiments,^29–31 and even the optical characteristics were investigated for TCI Pb_1−xSn_xSe nanoplates.^22,32 These outstanding investigations may substantially influence the development of the optical properties of topological materials. Therefore, the studies on plasmons in TCIs with nanostructures could offer a fresh look at the combined research between plasmonics and topological materials.

In this work, we employ TCI SnTe with nanostructured patterns for our plasmonic studies. Considering the insulating bulk surrounded by metallic surface states analogous to a standard core–shell-like structure in plasmonics,^17,33 we calculate, based on the finite-difference time domain (FDTD) method and density functional theory (DFT), the extinction spectra of SnTe nanopatterns. The localized field enhancements can be obtained in the visible-near-infrared (vis-NIR) spectral region. The numerical simulations are in tune with our theoretical analyses. Also, we explore the plasmon resonance peaks by calculating field distributions, and find that all resonances result from the interactions between cavity resonances and surface plasmons. Furthermore, the influence of grating height on resonance wavelengths shows that resonance peaks appear periodically, which could be attributed to the cavity resonances at some special height values. Our findings may benefit the development of plasmon-based biomedical sensors and integrated circuits in nanostructured topological materials.

Results and discussion

In order to get extra momentum,³⁴ SnTe nanogratings with a quartz substrate²⁵ are simulated for the observation of plasmons in this work. The schematic is shown in Fig. 1, where a presents the grating height, and b is the grating width which is just half of the period in the x direction with b = c = 15 nm. The structure parameter e = 15 nm describes the thickness between the two metallic surface layers. The thickness of the metallic surface states (d = 3 nm (ref. 35)) is defined as their penetration depth. The dielectric material surrounding the SnTe gratings has a refractive index of n = 2.0. Normal incident light propagates from the bottom of the structure along the z direction with x-polarization.

Fig. 1 (a) Schematic of the studied system: SnTe nanogratings embedded in a medium with a refractive index n = 2.0 on a quartz substrate. (b) Side view of the studied structure: SnTe nanogratings with height a, width b (15 nm), distance between adjacent gratings c (15 nm), penetration depth of SnTe metallic surface states d (3 nm),³⁵ and structure parameter e (15 nm) describing the thickness between the two metallic surface layers.

The extinction spectra of the SnTe nanogratings in our simulations are computed using the FDTD method³⁶ after calculating the permittivity ε(ω) using density functional theory. In FDTD simulations, to avoid non-physical reflections, convolution perfectly matched layer (CPML) as an absorbing boundary condition is applied at the lower and the upper boundaries in the computational window.^36–38 Periodic boundary conditions are taken in the x and y directions. The spatial discretization is defined as Δx = Δy = Δz = Δ = 0.1 nm, and temporal discretization is described by Δt = Δ/(2c₀), where c₀ is the velocity of light in a vacuum. The metallic surface states of SnTe are described using the Drude-critical point model, and the bulk state is presented by the critical point model.^39,40

In DFT calculations, the projector augmented wave method⁴¹ and the Perdew–Wang-91 (ref. 42) generalized gradient approximation of the exchange-correlation functional are used as implemented in Vienna Ab-initio Simulation Package (VASP),^43,44 and the spin–orbit coupling is included. The imaginary part of the permittivity ε(ω) is calculated via a summation over the empty states,⁴⁵ the real part of ε(ω) is obtained by the Kramers–Kronig transformation, and the complex shift⁴⁵ is set to be 0.004 in our simulations. To verify the calculated ε, the reflectivity derived by is compared with that obtained in the experiment⁴⁶ in Fig. 2. They are in good agreement when the energy is lower than 5 eV.

Fig. 2 Reflectivity of SnTe in our simulations (the red line) is in good agreement with the one given in the experiment (the black line)⁴⁶ when the energy is lower than 5 eV.

The resonance positions of the SnTe nanogratings are observed by calculating the extinction spectra. Firstly, the grating height is fixed as a = 8 nm, other structure parameters are described in the numerical method part. Extinction spectra are given using a solid blue line in Fig. 3(a). Then, four resonance peaks are observed at R₁ = 585 nm, R₂ = 800 nm, R₃ = 1263 nm and R₄ = 3084 nm.

Fig. 3 (a) Extinction spectra of the SnTe gratings when the grating height varies from 6 nm to 9 nm with an interval of 1 nm. The field distributions of the SnTe nanogratings with grating height (b) a = 8 nm and (c) a = 9 nm. In (b), the resonances are at 585 nm, 800 nm, 1263 nm and 3084 nm. In (c), the resonances are at 537 nm, 685 nm, 998 nm and 1839 nm.

The plasmon resonances of the TCI SnTe nanograting excited by a normal incident wave, just like those in graphene nanopatterns, approximately satisfy a ∼ mλ_sp/2, where m describes the number of half wavelengths which fit within the grating height for a certain mode, and λ_sp represents the surface plasmon wavelength.^47,48 In the considered spectral range, the intra-band Drude-like term plays an important role in the conductance, so the surface plasmon wave-number Re(k_sp) ≃ ℏω²/(2c₀γμ),⁴⁹ where ω is the angular frequency, μ is the chemical potential, and γ is the fine-structure constant, . In the definition of γ, e₀ is the electron charge, ℏ is the reduced Planck constant, ν_F is the Fermi velocity, g_s is the spin degeneracies, g_v is the valley degeneracies, and ε_avg is the average of the dielectric constants of media exposed to two sides of the metallic surface states. In our case of TCI SnTe with a (001) surface, we have ν_F = 2.3447 × 10⁵ m s⁻¹, g_s = 1 and g_v = 4. Then, for m = 1, we have the resonance wavelength . It predicts that the resonance wavelength will undergo a red shift if the structure parameter a increases. The resonance wavelength should happen at around λ_res ≈ 1181.6 nm when a = 8 nm, which is in the same spectral region as the results in our simulations.

When the incident wave interferes with the reflective wave inside the SnTe nanogratings, the cavity resonance wavelength can be presented as the formula ⁵⁰ where n₀ is the refractive index of bulk SnTe, and m₁ and m₂ are integers which characterize the cavity modes. The formula implies that λ will increase when a becomes larger and other conditions are fixed. For m₁ = 1, m₂ = 1, a = 8 nm, b = 15 nm, and n₀ = 44.721,⁵¹ we get the cavity resonance that is around λ ≈ 631.36 nm. It shows that cavity resonance will appear when the external exciting wavelength is at around 631.36 nm.

Based on the estimates of the surface plasmons and the cavity resonances above, we predict that the four resonances observed using a grating height fixed at 8 nm in Fig. 3(a) result from the interactions of surface plasmons and cavity resonances. In order to deeply explore the four resonance peaks, electric field distributions are plotted for the resonances at 585 nm, 800 nm, 1263 nm and 3084 nm. They are shown in Fig. 3(b), where the corners on the side-view of the structure in the x–z plane are defined as corner 1 (C1), corner 2 (C2), corner 3 (C3) and corner 4 (C4).

The field distribution at the resonance of 585 nm is dominantly composed of two parts in Fig. 3(b). One part is the field enhancements at C1 and C2 resulting from the surface plasmon, the other part is the cavity resonances inside the SnTe structure. Subsequently, the field distribution at the resonance of 800 nm is observed. The enhancements at C1 and C2 and the coupling between them are obvious. Meanwhile, the field interferences inside the structures become weaker than those observed at the resonance of 585 nm. And then, at the resonance of 1263 nm, field enhancements at C3 and C4 appear, and the couplings of field enhancements at adjacent corners (i.e. C1 to C2, C1 to C3, C2 to C4, and C3 to C4), contribute to the field distributions as well. But the enhancements at C1 and C2 still dominate the resonance. Finally, at the resonance of 3084 nm, we can observe that field enhancements happen at all of the corners, while cavity resonances nearly vanish. These results are in accordance with our previous prediction. Similarly, in silver nanocubes, field enhancements at the corners are studied as well,⁵² and applications are proposed in optoelectronic devices.⁵³

Up to now, we have shown that the cavity resonance dominates the field enhancement at the resonance of 585 nm. When the resonance wavelength increases, e.g. up to 3084 nm, the cavity resonance becomes very weak, and thus surface plasmon resonances play an important role at higher resonances. At the same time, the longer the resonance wavelength is, the greater the contribution the bottom corners (C3 and C4) make to field enhancements.

In order to explore the influence of the grating height on the resonances, we have modified the grating height from 6 nm to 9 nm with intervals of 1 nm. Extinction spectra are given in Fig. 3(a), and four resonances are observed for each grating height. Fig. 3(a) demonstrates that the resonances emerge nearly at the same wavelengths for a = 6 nm and a = 9 nm. Meanwhile for a = 7 nm and a = 8 nm, resonances appear around another different four wavelengths. These resonance peaks at different wavelengths form two series of resonances, in other words, there are two series of resonances for different grating heights, and each series includes four resonance peaks. The peaks occurring at the longer wavelengths are defined as the first series of resonances (SR1), and the peaks at the shorter wavelengths are defined as the second ones (SR2) as depicted in Fig. 3(a).

For SR2, field distributions at a = 9 nm are given as an example in Fig. 3(c) at resonances of 537 nm, 685 nm, 998 nm and 1839 nm. Here, at the low wavelengths (537 nm and 685 nm), the corners of the top surface dominate the field distributions. At the long wavelengths (998 nm and 1839 nm), the corners of the bottom surface come to contribute more to the field distributions. Compared with those at a = 8 nm, we can infer that the grating height affects the contributions of fields inside the structure, and further influences resonance peaks.

To further investigate the two series of resonances, we simulate the SnTe nanostructures with more fine grating heights ranging from 5 nm to 13 nm with intervals of 0.1 nm. Resonance peaks are given as a function of the grating height a in Fig. 4(a). When a varies from 5.2 nm to 5.4 nm, the resonance peaks have some sudden and remarkable shifts. In detail, for grating height a = 5.2 nm, the spectra show the third resonance peak (R3) at around 1263 nm, while for a = 5.4 nm, the spectra show R3 at around 998 nm. Other resonances (R1, R2 and R4) also exhibit sudden shifts at the two grating heights. Therefore, a = 5.3 nm is a transition point from SR1 to SR2. Similar things happen when a varies from 8.2 nm to 8.4 nm, from 6.4 nm to 6.6 nm and from 9.7 nm to 9.9 nm, among which the last two transitions are from SR2 to SR1.

Fig. 4 (a) Resonance peaks of SnTe nanopatterns with fine grating height a varying from 5 nm to 13 nm with 0.1 nm intervals. (b) Extinction spectra of the SnTe gratings with height a varying from 7.1 nm to 8.0 nm with intervals of 0.1 nm.

In Fig. 4(a), we also clearly see that the resonance positions vary from SR1 to SR2 then return to SR1 as a function of grating height, where the height variation is defined as the period (P), P = 3 nm in our simulations. Then, we give a relationship between the grating height (a), period (P) and resonances series in eqn (1),

where N is a positive integer. Eqn (1) indicates that if we have the grating height a, we can easily predict which series of resonances the spectra of the nanostructure belong to. More importantly, similar spectra for SnTe nanopatterns are observed in the same resonance series with different grating heights. For instance, when a changes from 7.1 nm to 8 nm, as shown in Fig. 4(b), no obvious shift happens. The insensitivity of the same resonance series with respect to the grating height could benefit the design and fabrication of nanodevices. Moreover, the sharp peaks, observed in Fig. 4(b), are an important benefit for precise detection and in biosensors. The period in Fig. 4(a) could be caused by cavity resonances inside the structure, as described in Fig. 3(b) and in Fig. 3(c). Finally, in the same resonance series, resonance peaks are red shifted with the increase of structure parameter a, which is shown by a dashed green line in Fig. 4(a). This is in good agreement with our previous theoretical predictions and in concordance with the phenomenon of graphene plasmons.^49,54

In addition, we investigate the influence of SnTe nanostructure parameter e on the extinction spectra. The parameter e, for instance, is selected to be 9 nm, 12 nm, 15 nm, 18 nm and 21 nm. Other structure parameters are fixed as a = 7.5 nm, b = c = 15 nm, and d = 3 nm. The extinctions are given in Fig. 5. The results demonstrate that strong resonances happen when e = 15 nm and 18 nm, while the resonances are weak for the other e values. This could be caused by the evolution between destructive interference and constructive interference due to the modification of structure parameter e.

Fig. 5 Extinction spectra of the SnTe nanogratings with structure parameter e = 9 nm, 12 nm, 15 nm, 18 nm and 21 nm. The other parameters are kept constant as a = 7.5 nm, b = c = 15 nm, and d = 3 nm.

TCI SnTe nanopatterns herein can be considered as a sample with the insulating bulk surrounded by metallic surface states. No matter what patterns are fabricated at the surface, the surface states can robustly exist even with defects or nonmagnetic impurities in the samples, as long as the crystalline (mirror) symmetry is preserved. Furthermore, the surface states can be modulated by external electric fields and strain effects.^18,55–57 Therefore, compared to the noble metals, high tunability could be accomplished for the plasmons in TCI SnTe nanopatterns, which may open a door to the development of tunable plasmonic nanodevices in topological materials.

Plasmons excited by massless Dirac fermions mainly focus on the THz domain.¹ But, rather recently, after intense efforts, the plasmon resonances of graphene have been tuned to the visible region using complicated patterns with micrometer order⁵⁸ and plasmons in the ultraviolet and visible range are excited by the TI nanostructures.¹⁰ In our case, plasmons of the TCI SnTe with nanogratings range from the visible to near infrared region, which has advantages for the detection of sensitive molecules and may benefit biomedicine.

Topological material nanostructures, such as Bi₂Se₃ with different nanopatterns, have been fabricated in experiments.²⁹ And TCI SnTe with the metallic Dirac surface band is observed using angle-resolved photoemission spectroscopy measurements.²⁵ Single crystalline SnTe nanostructures with controlled facets and morphologies are also synthesized successfully.³⁰ Our work could help to enhance the theoretical and experimental investigations on plasmons of topological materials with nanostructures.

Conclusions

In conclusion, the extinction spectra of TCI SnTe nanogratings, grown on quartz and embedded in a dielectric material with a refractive index of n = 2.0, are calculated by combining the FDTD and DFT methods, which can efficiently predict the plasmonic characteristics of the TCI system. Four resonance peaks in the vis-NIR spectral region are achieved using numerical calculations, and are also in good agreement with our theoretical analyses. We further show that all of the resonances are the interactions between cavity resonances and surface plasmons. Moreover, two series of resonances are observed by modifying the grating heights, where sudden shifts of resonance peaks appear with a period of 3 nm at some special heights described by eqn (1). In the same resonance series with different grating heights, TCI SnTe nanostructures exhibit similar plasmonic characteristics, which could benefit sample preparation and configuration design. The sharp resonance peaks, observed at different grating heights, e.g. a = 7.9 nm, pave an easy way to the application of TCI SnTe in biosensors. Potentially, the high tunability of TCIs also offers a possibility for making plasmonic switching nanodevices. Therefore, the TCI SnTe nanopatterns with insulating bulk states and metallic surface states, as a kind of intrinsic core–shell-like nanostructure, may establish a versatile platform for rich plasmonic physics and applications.

Acknowledgements

The authors thank Yizhou Liu, Mei Zhou, Chong Wang and Yong Xu for helpful discussion. The work is supported by the China Postdoctoral Science Foundation (Grants No. 2016M591156 and No. 2016M591123), the Ministry of Science and Technology of China, and the National Natural Science Foundation of China (Grants No. 11334006, No. 11204154, and No. 11574016). L. Z. is thankful for support from Open Research Fund Program of the State Key Laboratory of Low-Dimensional Quantum Physics (Grant No. KF201408). The calculations were done on the “Explorer 100” cluster system of Tsinghua University.

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