Broadband Optical Properties of Bi2Se3

Materials with high optical constants are of paramount importance for efficient light manipulation in nanophotonics applications. Recent advances in materials science have revealed that van der Waals (vdW) materials have large optical responses owing to strong in-plane covalent bonding and weak out-of-plane vdW interactions. However, the optical constants of vdW materials depend on numerous factors, e.g., synthesis and transfer method. Here, we demonstrate that in a broad spectral range (290–3300 nm) the refractive index n and the extinction coefficient k of Bi2Se3 are almost independent of synthesis technology, with only a ~10% difference in n and k between synthesis approaches, unlike other vdW materials, such as MoS2, which has a ~60% difference between synthesis approaches. As a practical demonstration, we showed, using the examples of biosensors and therapeutic nanoparticles, that this slight difference in optical constants results in reproducible efficiency in Bi2Se3-based photonic devices.


Introduction
High-refractive-index materials are the core of modern nanophotonics [1,2]. In particular, the real part n of the complex refractive index n determines the photonic devices' footprint, which scales as λ/n, where λ is a free space wavelength of light [3,4]. Moreover, the device performance is also highly dependent on n [5]. Hence, even a minor increase in the refractive index has a significant impact on nanophotonics [1]. However, classical high-refractive-index materials (TiO 2 , GaP, Si, and Ge) offer a limited range of available refractive indices, and these typically fall below 4 [2,[6][7][8]. It is thus necessary to search for new optical materials with large optical responses [1].
The appearance of graphene and other two-dimensional (2D) and layered materials [9][10][11] has led to the development of a novel materials platform, usually referred to as van der Waals (vdW) materials [12,13]. To date, there are more than 5000 potential vdW crystal structures which provide diverse optical responses [14] and can be employed together with or fully independently of bulk materials. One of the most promising is a family of transition broad spectral interval from 290 nm to 3300 nm. For the ellipsometry spectra analysis, we employed WVASE software, provided by the producer. We described the Bi 2 Se 3 sample using a four-layer model: roughness layer, Bi 2 Se 3 layer, silicon dioxide (SiO 2 ), and silicon (Si) substrate. The thicknesses of the layers were 21.7 nm (for the roughness layer), 44.4 nm (for the Bi 2 Se 3 layer), 265.7 nm (for the SiO 2 layer), and a semi-infinite layer for the silicon. In order to account for the surface roughness, we followed a standard approach [45] which involved introducing the Bruggeman effective medium approximation layer with equal fractions of air and investigating the Bi 2 Se 3 material. In addition, despite the uniaxial anisotropy of the Bi 2 Se 3 [12], we used an isotropic model for the Bi 2 Se 3 since ellipsometry is almost insensitive to the out-of-plane component because of the high in-plane refractive index, which significantly decreases the interaction between the light electric field and the out-of-plane dielectric function of the film.

Reflectance Measurements
The reflectance spectrum (λ = 450-950 nm) of the Bi 2 Se 3 sample was measured using a Biolam M-1 optical microscope (LOMO, Saint Petersburg, Russia) equipped with a 24 V, 100 W halogen light source and a QE65000 spectrometer (Ocean Optics, Dunedin, FL, USA).

First-Principle Calculations
The optical constants of the Bi 2 Se 3 were calculated using the GW@DFT approach implemented in the VASP package [46]. First, the atomic positions of the crystal (a = b = 0.4143 nm and c = 2.8636 nm) [47] were relaxed until the converged interatomic forces were less than 10 −2 eV/nm, and the unit cell was kept fixed. Next, we obtained groundstate one-electron wavefunctions using the density functional theory (DFT) and used them to initialize the GW routines. Finally, we calculated the imaginary and real parts of the frequency-dependent dielectric function within the GW approximation and derived the refractive indices and extinction coefficients of the material. The cutoff energy for the planewave basis set was 500 eV, while the first Brillouin zone was sampled with a Γ-centered 18 × 18 × 3 grid. The exchange correlation effects were described with a generalized gradient approximation (Perdew-Burke-Ernzerhof functional) [48], and the behavior of the wavefunctions in the core region was reconstructed with a projector augmented wave pseudopotentials [49].

Optical Visualization
The optical images of the Bi 2 Se 3 samples were recorded using an optical microscope (Nikon LV150L, Tokyo, Japan).

Atomic-Force Microscopy
The surface topography of the Bi 2 Se 3 films was examined using an atomic force microscope (NT-MDT Ntegra) operated in a semi-contact mode. AFM scanning was performed in air using HA_NC ETALON silicon tips (TipsNano, Tallinn, Estonia) with a tip-tapping resonant frequency of around 140 kHz and a spring constant of 3.5 N/m. The quantitative analysis was carried out using Gwyddion software (www.gwyddion.net; accessed on 1 January 2022).

X-ray Photoelectron Spectroscopy
The chemical state and composition were analyzed via X-ray photoelectron spectroscopy (XPS) using a Theta Probe spectrometer under high-vacuum conditions (base pressure < 2 × 10 −9 mbar) with a monochromatic Al-Kα X-ray source (1486.6 eV). The photoelectron spectra were acquired using the fixed analyzer transmission (FAT) mode with 50 eV pass energy. The spectrometer energy scale was calibrated on the Au4f 7/2 line (84.0 eV). The XPS spectra were acquired using charge-compensation under the pressure of~10 −7 mbar to avoid sample charging. For the elemental composition XPS analysis, Scofield's Factors were employed in the calculations.

Sample Characterization
Bi 2 Se 3 has a rhombohedral phase crystal structure with quintuple layers (Figure 1a). Our Bi 2 Se 3 thin film was prepared using the chemical vapor deposition (CVD) method and, therefore, had a uniform substrate coverage, confirmed by optical microscopy (Figure 1b). From the optical image in Figure 1b, one can notice that the synthesized sample has a roughness, similar to other Bi 2 Se 3 samples grown using CVD [50,51]. In order to obtain a qualitative estimate of the roughness, we measured the sample surface with an atomicforce microscope (Figure 1c). The AFM image yielded a root mean square roughness of 26 nm. After that, we verified the stoichiometry of the samples using X-ray photoemission spectroscopy (XPS) (Figure 1d,e). The XPS signal in Figure 1d,e shows a rich spectrum with Biand Se-related peaks, and a quantitative XPS analysis based on Bi4f and Se3d showed that the stoichiometry of the film reached ∼43%:57%, close to the expected 40%:60% (2 Bi: 3 Se) [52][53][54][55]. According to the previous studies of Bi 2 Se 3 , the peak Se 0 (Figure 1e) indicates elemental selenium (Se) [55][56][57]. Additionally, we performed Raman spectroscopy (Figure 1f,g) at two excitation wavelengths: 532 nm ( Figure 1f) and 632.8 nm (Figure 1g). Both spectra have two pronounced Raman peaks at 131.3 cm −1 and 174.3 cm −1 for λ = 532 nm and 131.9 cm −1 and 174.8 cm −1 for λ = 632.8 nm (Figure 1f,g). A comparison with Bi 2 Se 3 data from the literature allowed us to assign the first peak with the E 2 g phonon mode and the second peak with the A 2 1g phonon mode, and their positions are very close to those reported for Bi 2 Se 3 ( A 2 1g ∼131 cm −1 and E 2 g ∼174 cm −1 ) with a thickness above 20 nm [58]. In addition, we performed scanning electron microscopy (SEM) and X-ray diffraction (XRD), as is shown in Figure A2. The SEM image in Figure A2a confirms the morphology of our sample surface as observed using atomic-force microscopy (Figure 1c), and the XRD pattern ( Figure A2b [59]. In addition, we would like to note that the XPS spectra (Figure 1d,e) demonstrate the slight oxidation of the sample. However, other techniques, such as Raman spectroscopy (Figure 1f,g) and XRD ( Figure A2b), show only the presence of Bi 2 Se 3 , which additionally confirms that the sample was only slightly oxidized. Hence, this preliminary sample characterization confirmed that our sample was Bi 2 Se 3 and gives additional information about its roughness, which should be about 26 nm, and its thickness, which should be more than 20 nm.  . Black, red, green, purple, blue, orange, violet, and cyan colors label experimental, total, Se3p1/2 or Se3p3/2, Bi4f (BiOx), Bi4f5/2 or Bi4f7/2, Se3d3/2, Se3d5/2, and Se 0 XPS signals, respectively. Raman spectra of Bi2Se3 for (f) = 532 nm and (g) = 632.8 nm.

Spectrocopic Ellipsometry of Bi 2 Se 3
To obtain the broadband optical properties of the Bi 2 Se 3 , we measured the spectroscopic ellipsometry of our sample at several incident angles (θ = 50-70 • ) in a broad wavelength range (λ = 290-3300 nm). The resulting spectra of ellipsometric parameters Ψ and ∆ are plotted in Figure 2a,b. Since ellipsometry is a very accurate technique [60] that "feels" a system's nonidealities, we included in the optical model an effective medium approximation (EMA) layer [45,61] on top of the Bi 2 Se 3 film to account for the surface roughness (Figure 1b,c). We also assumed a negligible optical response from the Bi 2 Se 3 oxide and the surface conductive layer, and therefore did not include it in the optical model. Our ellipsometry analysis started with a point-by-point conversion approach [62]. We then used the results from the first step for the Lorentz oscillator description of the optical constants of the Bi 2 Se 3 (Figure 2c,d). We would like to note that unlike those of other semiconducting layered materials, the Bi 2 Se 3 optical response is better described via Lorentz oscillators than via Tauc-Lorentz oscillators [63] since Bi 2 Se 3 is a narrow bandgap (E g ≈ 0.3 eV) semiconductor [34]. In other words, the lowest energy incident photon has 0.376 eV energy, which is much larger than the bandgap E g = 0.3 eV of Bi 2 Se 3 . We also confirmed this bandgap value with band structure computations (the inset in Figure 2c) using density functional theory. Furthermore, to validate the Bi 2 Se 3 optical constants in Figure 2c,d and their predictive capabilities, we recorded the reflectance spectrum of our sample (the inset in Figure 2d) and compared it with the transfer matrix calculations [64] based on the refractive indices and extinction coefficients presented in Figure 2c,d. Therefore, our assumptions concerning a negligible optical response from the oxide layer and the EMA approach for roughness are valid because they were double-checked against the first-principle calculations and reflectance measurements, and there was good agreement between the AFM roughness of 26 nm and the effective ellipsometry roughness of 21.7 nm.   Table  A1.
In addition to this, we compared the resulting dielectric function of our CVD-grown sample with the first-principle calculations and dielectric function of molecular beam epitaxy (MBE)-grown Bi2Se3, reported by Fang and colleagues [40]. Interestingly, our recent publications [12,65] have shown that DFT dielectric function coincides with the optical response of monocrystals (an almost perfect match for n and a qualitative match with k), which in the case of layered materials are usually prepared using the exfoliation technique [66]. Hence, we can safely assume that DFT optical constants correspond with exfoliated Bi2Se3. In this case, the perfect match between the CVD-grown, exfoliated, and MBE-grown Bi2Se3 (Figure 2c,d) implies that the Bi2Se3 optical response is almost synthesis-invariant, unlike those of other vdW materials [23]. This property makes Bi2Se3 a promising vdW material for commercial use because its optical properties are reproducible.  Table A1.
In addition to this, we compared the resulting dielectric function of our CVD-grown sample with the first-principle calculations and dielectric function of molecular beam epitaxy (MBE)-grown Bi 2 Se 3 , reported by Fang and colleagues [40]. Interestingly, our recent publications [12,65] have shown that DFT dielectric function coincides with the optical response of monocrystals (an almost perfect match for n and a qualitative match with k), which in the case of layered materials are usually prepared using the exfoliation technique [66]. Hence, we can safely assume that DFT optical constants correspond with exfoliated Bi 2 Se 3 . In this case, the perfect match between the CVD-grown, exfoliated, and MBE-grown Bi 2 Se 3 (Figure 2c,d) implies that the Bi 2 Se 3 optical response is almost synthesis-invariant, unlike those of other vdW materials [23]. This property makes Bi 2 Se 3 a promising vdW material for commercial use because its optical properties are reproducible.
In addition to CVD and MBE technologies, Bi 2 Se 3 can also be synthesized using numerous other methods, e.g., solvothermal [67] and sonochemical [68] methods and mechanical exfoliation [69]. Unfortunately, for most synthesis methods, it is hard to find optical constants for comparison because researchers now focus primarily on the electronic properties of topological states rather than the optics of Bi 2 Se 3 . Nevertheless, the CVD and MBE methods are the most popular and well-developed for synthesizing two-dimensional and layered materials and, therefore, the most important for the scientific community. Additionally, we provide first-principle computations, which give optical constants close to those of the exfoliated samples [12]. Hence, we can conclude that first-principle computations yield optical constants for exfoliated Bi 2 Se 3 , expanding our comparison to the three synthesis methods (CVD, MBE, and exfoliated) and confirming the synthesis-independent optical response of Bi 2 Se 3 .

Applications of Bi 2 Se 3
To demonstrate the invariant performance of Bi 2 Se 3 -based photonic devices, we chose two applications: a surface plasmon resonance (SPR) biosensor [70] and the heating of nanoparticles for cancer treatment [21]. Performance-invariance is imperative for the reliable industrial implementation of Bi 2 Se 3 .
We commence with a Bi 2 Se 3 -based SPR biosensor. In a common approach [70] to SPR-sensitivity enhancement, one usually deposits vdW materials on top of gold (or other plasmonic material) in a biosensor using the Kretschmann scheme (the inset in Figure 3b) [71]. The benefit of the added vdW material is twofold: (i) it increases the sensitivity of the biosensor; (ii) it enhances the immobilization efficiency of the detected molecules. Since Bi 2 Se 3 is a topological insulator, one might expect that Bi 2 Se 3 could also give a plasmonic response. This expectation is correct, and plasmonic modes in Bi 2 Se 3 were observed in the THz range [39,72]. Above the bandgap, the optical response from the topological states is combined with interband transitions in bulk material, with the weight of the former decreasing with the increase in the thickness of the material [40,73]. At the same time, topological insulators can support guided surface electromagnetic waves, provided that the real part of the permittivity is negative [31,74]. In the case of Bi 2 Se 3 , at a standard SPR wavelength of λ = 635 nm (E = 1.953 eV), the real part of the dielectric permittivity is positive, and thus we do not foresee its application as a replacement for plasmonic metal. Assuming that Bi 2 Se 3 is an auxiliary layer, the optical constants in Figure 2c,d allow the estimation of the sensitivity enhancement. Using the transfer matrix calculations, we determined the dependence of the reflection coefficient on the angle of incidence (Figure 3a) and the biosensor sensitivity (Figure 3b) for the CVD and MBE-grown Bi 2 Se 3 . The close characteristics (Figure 3a,b) of the biosensors for both the CVD and MBE Bi 2 Se 3 show that the device performance remains almost independent of the choice of synthesis method. During the calculations, we neglected the contribution of the topological states to the optical response (see Figure A3 for the estimation of the error introduced by this oversight). Given the recent success of the fabrication of nanospheres from vdW materials [21,75], we also considered Bi2Se3 nanospheres for efficient heating in the therapeutic window, known as NIR-I (700-980 nm) [21]. To demonstrate the efficiency invariance of Bi2Se3 for the optical response of nanoparticles, we employed the Mie theory [76] to calculate the multipole decomposition of the extinction spectrum (Figure 3c Like the biosensor, the heating efficiency of the Bi2Se3 nanospheres was very close for CVD and MBE-grown Bi2Se3, especially in the practically important NIR-I spectral region. Thus, the synthesis-independent optical constants of Bi2Se3 lead to the synthesis-independent performance of Bi2Se3 optical devices. In addition, it is worth comparing the performance of Bi2Se3-based devices with the performance of devices made from other materials. For the SPR comparison, we included the performance of graphene (Gr) [62] and MoS2 [77] in Figure 2b. This shows that despite the enormously high refractive index of Bi2Se3 (n~5.3) at a standard SPR wavelength of λ = 635 nm, Bi2Se3 demonstrates slightly less SPR sensitivity than graphene and MoS2 owing to its strong optical absorption (k~3.3). In addition, large optical constants make Bi2Se3 a suitable material for the heating of NPs for cancer treatment. Indeed, the comparison of Bi2Se3 with traditional materials such as Au [78], Si [79], and MoS2 [21] in the therapeutic window NIR-I reveals a more than tenfold enhancement in heating efficiency (Figure 3f). Given the recent success of the fabrication of nanospheres from vdW materials [21,75], we also considered Bi 2 Se 3 nanospheres for efficient heating in the therapeutic window, known as NIR-I (700-980 nm) [21]. To demonstrate the efficiency invariance of Bi 2 Se 3 for the optical response of nanoparticles, we employed the Mie theory [76] to calculate the multipole decomposition of the extinction spectrum (Figure 3c), scattering (Figure 3d), and absorption (Figure 3e) cross-sections for a nanosphere with a standard diameter of d = 100 nm in a water environment using the dielectric function of CVD and MBE-grown Bi 2 Se 3 . Using these cross-sections, we estimated the spectral dependence of the heating of Bi 2 Se 3 nanoparticles under constant laser irradiation (I 0 = 3.2·10 5 W/m 2 ). Like the biosensor, the heating efficiency of the Bi 2 Se 3 nanospheres was very close for CVD and MBE-grown Bi 2 Se 3 , especially in the practically important NIR-I spectral region. Thus, the synthesis-independent optical constants of Bi 2 Se 3 lead to the synthesis-independent performance of Bi 2 Se 3 optical devices.
In addition, it is worth comparing the performance of Bi 2 Se 3 -based devices with the performance of devices made from other materials. For the SPR comparison, we included the performance of graphene (Gr) [62] and MoS 2 [77] in Figure 2b. This shows that despite the enormously high refractive index of Bi 2 Se 3 (n~5.3) at a standard SPR wavelength of λ = 635 nm, Bi 2 Se 3 demonstrates slightly less SPR sensitivity than graphene and MoS 2 owing to its strong optical absorption (k~3.3). In addition, large optical constants make Bi 2 Se 3 a suitable material for the heating of NPs for cancer treatment. Indeed, the comparison of Bi 2 Se 3 with traditional materials such as Au [78], Si [79], and MoS 2 [21] in the therapeutic window NIR-I reveals a more than tenfold enhancement in heating efficiency (Figure 3f). Therefore, Bi 2 Se 3 is a promising material for absorbing and heating photonic applications thanks to its extraordinarily high optical response.

Conclusions
In summary, we have reported the broadband (290-3300 nm) optical properties of Bi 2 Se 3 , a typical representative of van der Waals (vdW) topological insulators. Our study shows that Bi 2 Se 3 has ultrawide absorption, with an extinction coefficient above 0.1, and an enormously large dielectric response, with a refractive index above 5. This was unambiguously verified using theoretical computations within the density functional theory framework and reflectance spectroscopy. More importantly, we found that Bi 2 Se 3 optical constants are synthesis-invariant, which is highly desirable for optical engineering. As a result, we envision Bi 2 Se 3 as an essential material in the next generation of nanophotonic nanostructures, useful in countless applications, including biosensing [70], theranostics [21], photodetection [80], light focusing [81], and superabsorbers [82]. Therefore, Bi2Se3 is a promising material for absorbing and heating photonic applications thanks to its extraordinarily high optical response.

Conclusions
In summary, we have reported the broadband (290-3300 nm) optical properties of Bi2Se3, a typical representative of van der Waals (vdW) topological insulators. Our study shows that Bi2Se3 has ultrawide absorption, with an extinction coefficient above 0.1, and an enormously large dielectric response, with a refractive index above 5. This was unambiguously verified using theoretical computations within the density functional theory framework and reflectance spectroscopy. More importantly, we found that Bi2Se3 optical constants are synthesis-invariant, which is highly desirable for optical engineering. As a result, we envision Bi2Se3 as an essential material in the next generation of nanophotonic nanostructures, useful in countless applications, including biosensing [70], theranostics [21], photodetection [80], light focusing [81], and superabsorbers [82].