An all-photonic isolator using atomically thin (2D) bismuth telluride (Bi₂Te₃)

Abstract

Here, we demonstrate two-dimensional (2D) Bi₂Te₃ with broadband Kerr nonlinear optical response, which can be used for nonreciprocal light propagation in passive photonic isolators. The self-induced diffraction patterns generated at various wavelengths (650 nm, 532 nm, and 405 nm) in the far field are investigated to calculate the nonlinear refractive index (n₂) and third-order nonlinear susceptibility χ⁽³⁾_total of the synthesized 2D Bi₂Te₃ using the SSPM (spatial self-phase modulation) spectroscopy method. 2D Bi₂Te₃ exhibits a significant nonlinear refractive index on the order of ≈10⁻⁴ cm² W⁻¹, which is higher than that of graphene. The laser-induced hole-coherence effect accounts for the significant magnitude of the third-order nonlinear susceptibility χ⁽³⁾_monolayer (on the order of 10⁻⁷ e.s.u.). Surface engineering is applied to realize a fast-response photonic system. Bader charge analysis (ab initio simulations) was performed to probe the interaction between 2D Bi₂Te₃ and different solvent molecules. 2D hBN exhibits reversible saturation, reducing the intensity of the propagating beam. Leveraging the enhanced Kerr nonlinearity of 2D Bi₂Te₃, a nonlinear photonic isolator (2D-Bi₂Te₃–2D-hBN heterostructure) that disrupts time-reversal symmetry has been successfully demonstrated, enabling unidirectional light propagation. This demonstration of the photonic isolator shows Bi₂Te₃ as a novel 2D material, expanding its potential applications across multiple photonic devices, including detectors, modulators, and switches.

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1. Introduction

The primary objective of designing photonic isolators is to provide nonreciprocal light transmission, with applications in optical telecommunications and integrated photonics.¹ The traditional methods for achieving optical nonreciprocity include acousto-optically induced interband photonic transitions,^2,3 optomechanical mechanisms,^4,5 customized waveguides,^6,7 micro-resonator structures,^1,8 and others. The most traditional method for achieving optical isolation is the Faraday effect, which relies on an applied magnetic field along the optical axis. An optical isolator, also known as an optical diode, is a component that allows unidirectional light passage. The functionality of traditional optical isolators depends on the Faraday effect, which arises from the magneto-optic effects in the primary component, the Faraday rotator. However, it suffers from certain disadvantages, such as a limited isolation ratio, temperature sensitivity, magnetic-field dependence, large size and weight, and wavelength dependence. Nonetheless, integrated isolators that do not depend on magnetism and have less temperature sensitivity and wavelength dependence have also been developed in recent years.

Over the past decade, the application of photonic diodes has remained relatively underexplored. Wu et al. reported a two-dimensional (2D) Te nanosheet (Ns)-based air-stable nonlinear photonic diode, attributed to its strong light–matter interaction in the visible-to-infrared band.⁹ Furthermore, the authors documented the distinct Kerr nonlinearity exhibited by 2D graphdiyne, which facilitated the development of a nonreciprocal light propagation device. Recently, there has been a significant increase in interest in the utilization of SSPM (spatial self-phase modulation) spectroscopy for probing non-reciprocal light propagation in 2D materials such as graphene, 2D Te, NbSe₂, MoS₂, NiTe₂, MoTe₂, MoSe₂, SnS, Ti₃C₂T_x MXene, antimonene, Ti₃CN, violet phosphorus nanosheets, Sb₂Se₃ nanosheets, and TaSe₂.^9–21 SSPM spectroscopy can measure the nonlinear refractive index coefficients of 2D materials. Studies show that the value of the nonlinear refractive index n₂ of graphene is estimated to be on the order of 10⁻⁵ cm² W⁻¹, as calculated using SSPM.²² Zhang et al. measured the nonlinear refractive index of black phosphorus to be in the range of 10⁻⁵ cm² W⁻¹, derived using the SSPM spectroscopy method.²³ Among various 2D materials, Bi₂Te₃ is known as a topological insulator, characterized by its unique energy band structure. The novel optical properties of topological insulators make them highly desirable for applications in nonlinear optics, light modulation, fiber lasers, and other related photonic fields.^24–26 A topological insulator (TI) is a novel kind of quantum matter characterized by a distinct bulk gap and an odd number of relativistic Dirac fermions present on its surface.^27–29 Unlike graphene, TIs possess distinct photonic and opto-electronic characteristics that arise from the combined influence of enhanced spin–orbit interactions and surface states shielded by time-reversal symmetry.³⁰ In addition to their distinct quantum properties, TIs exhibit a wide range of nonlinear optical (NLO) responses^31–34 that span from the visible to terahertz frequencies. This NLO response has been confirmed in numerous experimental setups, including diverse mode-locked or Q-switched laser operations conducted by various research groups.^35–39 Furthermore, the Z-scan approach has shown that the third-order nonlinear refractive index of the TI Bi₂Te₃ nano-platelets is on the order of 10⁻⁸ cm² W⁻¹.³⁴

This study introduces an innovative method for achieving nonreciprocal light propagation in a nonlinear photonic isolator by utilizing the enhanced nonlinear optical response of 2D Bi₂Te₃ and the reverse saturable absorption behavior of 2D hBN. 2D Bi₂Te₃, an ultrathin TI, has been successfully synthesized via liquid-phase exfoliation. Our work indicates that 2D Bi₂Te₃ exhibits a narrow electronic bandgap, enabling broadband nonlinear optical response upon the passage of a laser beam. The nonlinear optical characteristics and their coefficients, specifically n₂ (nonlinear refractive index) and χ⁽³⁾_total (third-order nonlinear susceptibility), were determined using the SSPM spectroscopic method. The findings demonstrate that 2D Bi₂Te₃ has a large nonlinear refractive index (≈10⁻⁴ cm² W⁻¹), indicating a strong nonlinear optical response. This suggests that 2D Bi₂Te₃ may serve as an exceptional optical material for photonic isolators and all-photonic devices.

The temporal evolution of the SSPM pattern was analyzed. The distortion in the SSPM pattern was also analyzed, and the refractive index variation was evaluated. Solvent exchange is utilized to identify the most efficient system capable of delivering rapid responses under laser beam exposure. The evolution of the SSPM pattern under different wavelengths and solvents was determined. Ab initio computer simulations were also performed to further investigate the interactions of 2D Bi₂Te₃ with solvents. A conclusion is drawn from the solvent–2D Bi₂Te₃ system, which yields a rapid response. 2D hBN exhibits reverse saturable absorption (RSA) with a large electronic bandgap of 5.28 eV; however, exciting the diffraction rings poses challenges. The coupling of the two materials to form a hybrid 2D-Bi₂Te₃/2D-hBN structure leads to propagational symmetry-breaking between the forward (2D-Bi₂Te₃/2D-hBN) and reverse (2D-hBN/2D-Bi₂Te₃) directions, facilitating the excitation of unidirectional diffraction rings. The results confirm that the suggested photonic isolator facilitates nonreciprocal light transmission in optical telecommunications and integrated photonics. In contrast to bulky traditional optical isolators, the proposed method utilizes the SSPM technique, leveraging 2D materials, specifically 2D Bi₂Te₃ and 2D hBN.

2. Results and discussion

2.1 Synthesis and characterization of the 2D Bi₂Te₃ nanostructure

The synthesis and characterization are discussed in section S1 of the SI.

2.2 Experimental section: SSPM setup

Fig. 1a depicts the method utilized to characterize the nonlinear optical response in a spatial self-phase modulation (SSPM) experiment. This experiment utilized three lasers with distinct continuous wavelengths of 650, 532, and 405 nm. The laser beams were directed through a 20 cm focal-length convex lens and onto a cuvette containing the 2D Bi₂Te₃ sample. The SSPM effect is observable as the laser beam traverses the cuvette, producing a diffraction pattern on the far-field screen, which is subsequently captured using a CCD camera. The underlying principle of the nonlinear optical characteristics of two-dimensional materials based on SSPM can be explained by the Kerr nonlinearity.

Fig. 1 Experimental setup and SSPM spectroscopy-based observation in the far field. (a) Diagram depicting the SSPM experimental setup. (b) Schematic representation of the interaction between 2D Bi₂Te₃ and light, showing the SSPM phenomenon arising from electronic coherence involvement with the laser beam. (c–e) Evolution of the diffraction pattern in the far field with varying intensities at different wavelengths (λ = 650, 532, and 405 nm). (f) Captured diffraction patterns in the far field at different wavelengths of 650, 532, and 405 nm at a constant concentration C₂ and a cuvette length of 10 mm. (g) Variation in the number of diffraction rings in response to changes in laser intensity (λ = 650 nm) at different concentrations, while maintaining a constant cuvette length of 10 mm. (h) Different cuvette lengths (L = 10 mm, 5 mm, and 1 mm) affect the number of diffraction rings with varying intensities at a constant wavelength (λ = 650 nm) and concentration C₂.

2.3 Basic understanding of the SSPM phenomenon and calculation of the nonlinear optical coefficients n₂, χ⁽³⁾_total and χ⁽³⁾_monolayer

This study explores the nonlinear Kerr effect, which is essential for understanding the nonlinear optical responses of 2D Bi₂Te₃. The Kerr nonlinear effect is defined in eqn (1), which establishes the correlation between the intensity of the incident laser beam (I) and the refractive index (n):

n = n₀ + n₂I (1)

The terms n₀ and n₂ represent the linear and nonlinear refractive indices of the material, respectively.⁴⁰

In the outgoing Gaussian light, there are at least two distinct locations, r₁ and r₂, where the slopes of the distribution curve, represented by and , are equal and have the same phase. This is evident from the Gaussian distribution of the nonlinear phase shift, as shown in Fig. 1b. Thus, the output light intensity profile exhibits a consistent phase difference while maintaining the same slope points. Clearly, these two points satisfy the requirement for interference. When Δ₀ ≥ 2π, diffraction rings appear. The self-induced diffraction pattern is observed as rings, which can be either bright or dark, in the far field.

The phase shift, which is crucial for the SSPM effect, generates a self-diffraction pattern in the far field, as shown in Fig. 1(b).

The nonlinear refractive index n₂ is expressed as

n₀ and n₂ represent the linear and nonlinear refractive indices of the material, respectively. is an important parameter for evaluating the nonlinear refractive index of a two-dimensional material, defined as the change in the number of rings with a change in the focused beam intensity of the incoming laser. L_eff is the effective transmission length of the laser beam inside the cuvette. The third-order nonlinear susceptibility χ⁽³⁾_total is employed to characterize the nonlinear optical characteristics of materials.^11,41,42 It can be written as

Here, c represents the speed of light in a vacuum, n₀ is the linear refractive index of the IPA solvent, and n₂ defines the effective length that the laser beam propagates through the cuvette. However, the effective number of 2D materials available in the cuvette has a direct effect on the value of χ⁽³⁾_total. Hence, it is necessary to determine the value of the third-order nonlinear susceptibility caused by a single layer of two-dimensional flakes χ⁽³⁾_monolayer. Detailed calculations of n₂ and χ⁽³⁾_total are provided in the SI section S2. The relationship between the overall electric field strength E_total and the electric field strength E_monolayer traveling through a single layer of Bi₂Te₃ can be mathematically represented as^22,43

In this context, N_eff represents the number of 2D Bi₂Te₃ layers in the solution through which the beam passes. The relationship between χ⁽³⁾_total and χ⁽³⁾_monolayer can be expressed as^11,14,43

χ⁽³⁾_total = N_eff²χ⁽³⁾_monolayer (5)

Wu et al.⁴⁴ described the SSPM phenomenon as originating from nonlocal and intraband AC electron coherence. Nearly all portions of the incoming laser beam propagation, and Rayleigh scattering is found to be low, the SSPM procedure is inherently a third-order nonlinear optical process. Additional experiments presented in the SI section S7 clarify that the SSPM phenomenon has an electronic origin rather than a thermal one.⁴⁵ The concentration of 2D Bi₂Te₃ is varied to quantify the light–matter interaction taking place, although 0.25 mg mL⁻¹ was selected for further experimental purposes.

The N_eff calculation is presented in the SI section S3. Fig. 1c(①–⑦), Fig. 1d(①–⑦), and Fig. 1e(①–⑦) show the SSPM diffraction patterns recorded on the far-field screen at wavelengths of 650, 532, and 405 nm, which were recorded using a CCD camera. A linear increase in the number of rings is observed as the intensity is increased. The increase in intensity leads to a corresponding increase in both the horizontal and vertical diameters of the diffraction pattern. Fig. 1f illustrates the relationship between intensity and the number of rings for laser beams with varying wavelengths (650, 532, and 405 nm). The values calculated from the curve fitting were found to be 1.26 cm² W⁻¹, 4.64 cm² W⁻¹, and 7.55 cm² W⁻¹ at λ = 650, 532, and 405 nm, respectively. It is concluded that as the wavelength decreases, the energy of photons increases, thereby intensifying the SSPM effect. A control experiment was performed in which the laser beam interacted with the solvent devoid of any 2D material, and no diffraction pattern was observed on the far-field screen. This experiment is described in section S9 of the SI. Hence, it was concluded that the pattern generation is due to the presence of 2D Bi₂Te_3. Additional parameters that also contribute to the effective formation of SSPM include the effective concentration of 2D material in the solvent and the effective path length of the incoming laser beam within the cuvette. Fig. 1g demonstrates the relationship between the intensity of laser light with a wavelength of 650 nm and the number of rings at varying concentrations of the 2D material active in the solution. This correlation is observed when analyzing various concentrations of suspended two-dimensional material in solution, as shown in Fig. 1g. A linear relationship between laser intensity and the number of diffraction rings is demonstrated using curve-fitting analysis. SSPM spectroscopy was conducted at specific concentrations of 0.0625 (C₀), 0.13 (C₁), and 0.25 (C₂) mg mL⁻¹, respectively. The corresponding slopes for were determined to be 1.03, 1.17, and 1.26 cm² W⁻¹. As the effective concentration of 2D materials increases in the solution, light–matter interactions increase, leading to more rings in the diffraction pattern. Increasing the value of for the corresponding order of the concentration values results in enhanced light–matter interaction. The variation in the number of rings in relation to the intensity of the incoming laser beam for various cuvette lengths is shown in Fig. 1h. It was observed that changes in the number of rings with respect to intensity increase with increasing cuvette length. The slope is calculated to be 0.83, 1.14, and 1.26 cm² W⁻¹ for cuvette thicknesses of 1 mm, 5 mm, and 10 mm at a constant concentration of 0.25 mg mL⁻¹ (C₂) and wavelength λ = 650 nm. Increasing the cuvette thickness increases light–matter interaction, leading to greater phase shifts and more rings in the diffraction pattern. As the effective propagation length through the cuvette decreases, light–matter interaction decreases, resulting in fewer diffraction pattern rings at a fixed concentration and constant intensity. The experiment revealed that the values of n₂ are 2.7 × 10⁻⁴, 8.14 × 10⁻⁴, and 10.1 × 10⁻⁴ cm² W⁻¹ at the specified wavelengths of 650, 532, and 405 nm. The third-order nonlinear susceptibility (χ⁽³⁾_total) is found to be 0.01569, 0.04741, and 0.05887 e.s.u., respectively, at wavelengths of 650, 532, and 405 nm. An increase in both cuvette thickness and concentration of suspended 2D Bi₂Te₃ within the cuvette leads to a corresponding increase in the number of rings per unit intensity. The computed values of n₂ and χ⁽³⁾_total obtained from the experiment described above are presented in Table 1. The values of n₂ and χ⁽³⁾_total increase significantly as the travel time within the cuvette decreases and the amount of active material increases. The measured values of n₂ and χ⁽³⁾_total for 2D Bi₂Te₃ obtained through the SSPM approach are determined to be much higher than those of other transition metal dichalcogenides (TMDCs) (compared in Table 1 and the SI Table S1).

Table 1 Experimental values of n₂ (nonlinear refractive index), χ⁽³⁾_total (third-order nonlinear susceptibility), and χ⁽³⁾_monolayer (third-order nonlinear susceptibility for a monolayer)

Wavelength (nm)	Concentration (mg mL^−1)	L (mm)	Solvent	dN/dI (cm^2 W^−1)	N eff	n 2 (cm^2 W^−1)	χ ^(3)total (e.s.u.)	χ ^(3)monolayer (e.s.u.)
650	0.25	10	NMP	1.26	361	2.7 × 10^−4	0.01569	1.2 × 10^−7
650	0.25	10	IPA	1.29	361	3.038 × 10^−4	0.01466	1.12 × 10^−7
532	0.25	10	NMP	4.64	361	8.14 × 10^−4	0.04741	3.63 × 10^−7
405	0.25	10	NMP	7.55	361	10.1 × 10^−4	0.05887	4.51 × 10^−7
650	0.13	10	NMP	1.17	188	2.51 × 10^−4	0.01456	4.12 × 10^−7
650	0.0625	10	NMP	1.07	90	2.22 × 10^−4	0.00129	1.59 × 10^−7
650	0.25	5	NMP	1.14	180	2.45 × 10^−4	0.01419	8.76 × 10^−7
650	0.25	1	NMP	0.83	36	0.00178	0.10332	7.97 × 10^−7

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The electron experiences minimal or zero resistance as it moves over the surface of 2D Bi₂Te₃. This property allows conduction even in the presence of many oxidized surface flaws in the TI material. The Fermi energy^46,47 of n-type Bi₂Te₃ is higher than that of intrinsic Bi₂Te₃ and p-type Bi₂Te₃. Furthermore, the saturation density of photoexcited carriers in p-type Bi₂Te₃ (n_p) exceeds that in intrinsic Bi₂Te₃ (n_i) and n-type Bi₂Te₃.

D(E) represents the density of states (DOS) at energy E. The function f(E) represents the probability distribution of carriers. E_photon is the energy of photon-excited electrons. E_Fn and E_Fp are the Fermi energies of the n-type and p-type Bi₂Te₃, respectively. The dopant type of transition metal (TI) changes the Fermi level without altering its energy bandgap. Thus, the photocarrier density and saturated photocarrier density vary between n-type and p-type materials. In this study, 2D Bi₂Te₃ is found to be p-type, which may in turn give rise to laser-induced hole-based AC coherence, as holes are the dominant charge carriers compared to electrons in p-type materials.

A large nonlinear optical coefficient is observed due to the laser-induced hole-coherence effect.⁴⁸ The identification of high-lying bulk states positioned above the initial bulk state was predicted to be the reason for this effect. However, here, the laser beam intensity is kept low. The significant magnitude of the nonlinear optical coefficient is attributable to a different factor. p-Type Bi₂Te₃ has a lower Fermi level than n-type Bi₂Te₃, resulting in a greater capacity to generate excited carriers. These hole-type carriers are involved in the laser-induced hole coherence that gives rise to high values of n₂, χ⁽³⁾_total, and χ⁽³⁾_monolayer.

2.4 Wind chime model: diffraction pattern generation in media with different viscosities

2.4.1 Time evolution under different wavelengths and variable intensities. In this analysis, we investigate the process of pattern formation and the fundamental mechanism of SSPM. Wu et al. introduced a model to elucidate the mechanism of pattern formation in the SSPM process, including the time required for the pattern to reach a consistent level of intensity.¹¹ This model suggests that at the initial interaction of the incoming laser beam with 2D Bi₂Te₃, the angle between 2D Bi₂Te₃ and the laser beam's electric field is arbitrary. This incoming laser beam polarizes the suspended 2D Bi₂Te₃ in the solution.⁴³ Eventually, the polarized 2D Bi₂Te₃ aligns along with the external electric field and the polarization direction of the incident laser's electric field due to the energy relaxation process.¹¹ As time progresses, the macroscopic angle between 2D Bi₂Te₃ decreases, leading to a more precise alignment with the electric field, which results in a continuous increase in the number of diffraction rings. Once all 2D Bi₂Te₃ in the path length of the incoming laser beam are perfectly aligned, the number of rings reaches its maximum value.

To validate this model and its dependence on the medium's viscosity, we performed an experiment to investigate the effect of the intense laser beam's polarisation on the suspended 2D Bi₂Te₃ in IPA and NMP. Fig. 2a explains the wind chime model for 2D Bi₂Te₃ in the NMP/IPA dispersion. At first, a point of outgoing Gaussian beam emerges in the far field, progressively transforming into a complete circular pattern of light diffraction when the suspended 2D Bi₂Te₃ aligns itself with the electric field of the incoming laser beam. Fig. 2b(①–⑥), Fig. 2c(①–⑥), and Fig. 2d(①–⑥) show the SSPM pattern's evolution under continuous-wave laser beams at λ = 650, 532, and 405 nm. The time required for the 2D Bi₂Te₃ domains to fully align with the electric field of the incident laser beam corresponds to the time required for the full formation of the diffraction ring pattern. An exponential model can explain the process of ring generation in time.¹³

N = N_max(1 − e^t/τ_rise) (6)

Fig. 2 The evolution of the far-field diffraction pattern over time. (a) Depiction of the wind chime model and the evolution of the SSPM pattern in the far field. (b–d) The far-field diffraction pattern over time at different wavelengths (λ = 650, 532, and 405 nm). (e–g) Temporal variation in the number of diffraction rings at various wavelengths (λ = 650, 532, and 405 nm).

Here, N is the quantity of rings observed in the diffraction pattern, N_max represents the maximum number of rings that are created when the laser intensity remains constant, and τ_rise is the duration required for pattern development.

The wind chime model properly represents the time (T) required for the full-diameter diffraction pattern to appear.^11,14

Here, ε_r represents the relative dielectric constant of 2D Bi₂Te₃, understood to be 5,⁴⁹ and η represents the viscosity coefficient of the solvent. The values are 2.4 × 10⁻³ Pa s at 20 °C and 1.65 × 10⁻³ Pa s at 25 °C for IPA and NMP, respectively, and ξ denotes the segment of the fluid sphere that is proximal to the 2D nanostructure. The variable R_C represents the radius of 2D Bi₂Te₃, specifically the radius of its exposed surface in contact with the solvent; h represents the vertical height of the suspended 2D nanostructure; and I denotes the intensity of the incident laser beam. The value of the domain radius (R_C) = the radius of the nanostructure (50 nm), obtained by atomic force microscopy, and the flake thickness (h) is 3.5 nm. Furthermore, the theoretically predicted values of T are 0.32 s, 0.478 s, and 0.263 s for wavelengths of 650, 532, and 405 nm, respectively, in NMP solvent. These values, obtained by curve fitting, are compatible with the experimentally derived values of 0.33 s, 0.53 s, and 0.30 s for wavelengths of 650, 532, and 405 nm, respectively, as shown in Fig. 2e, f and g. The number of rings is greater at 532 nm than at other wavelengths, indicating enhanced light–matter interaction; this results in a longer time for diffraction pattern formation. The time taken for the diffraction pattern to reach the maximum number of rings is shown in Table 2 for different wavelengths and viscous media.

Table 2 Experimental and theoretical values of the time required for the diffraction patterns to reach the maximum number of rings

Sample	Solvent	Wavelength [nm]	C [mg mL^−1]	L [mm]	η [mPa s]	τ rise [s]	Experimental T [s]	T from the wind chime model [s]
2D-Bi2Te3	NMP	650	0.25	10	1.65	0.1139	0.333	0.328
2D-Bi2Te3	IPA	650	0.25	10	2.4	0.213	0.5	0.4
2D-Bi2Te3	NMP	532	0.25	10	1.65	0.166	0.5333	0.6179
2D-Bi2Te3	NMP	405	0.25	10	1.65	0.106	0.3	0.263

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2.4.2 Time evolution in different viscous media. As discussed previously, we have examined nonlocal electron coherence in detail. In our sample, each flake-like nanostructure represents a domain. Initially, charge carriers, including photocarriers, excitons, free electron–hole pairs, and intrinsic electrons and holes, are distinctly out of phase, regardless of whether they exist in separate or in the same domains. Moreover, every domain possesses an arbitrary orientation.

Wu et al.¹¹ employed a wind chime model to describe the evolution of electron coherence in the nonlocal regions induced by SSPM. According to the authors, initially, there is no fixed orientation angle between a polarized 2D Bi₂Te₃ nanostructure and the electric field. The electric field induces an energy relaxation process, prompting the flakes or nanostructures to realign in such a manner that each domain takes an orientation parallel to the polarization of the external electric field. This configuration makes each domain appear suspended by a vertical filament. The aforementioned domain consists of the nanostructure and the fluid surrounding it. A depiction resembling a wind chime model is presented in Fig. 2a. Now, this laser beam exerts a torque on the flake/nanostructure, though the fluid's viscous force counteracts it. The viscous force experienced by the nanostructure through the boundary is found to be¹¹

In this context, Ω represents the rotational velocity, and ξ denotes the segment of the fluid sphere that is close to the disc. The viscous force is directly dependent upon the value of η (viscosity coefficient). Although these nanostructures have a quasi-two-dimensional structure with an exposed surface area, this creates non-passivated bonds on their surfaces. Thus, the probability of the interaction with a solvent molecule is high. In our experiment, IPA showed a higher viscosity coefficient than NMP. Hence, the viscous force is lower in NMP-2D Bi₂Te₃ than in IPA-2D Bi₂Te₃. The time required for the diffraction pattern to fully form IPA-2D Bi₂Te₃ is longer than that for NMP-2D Bi₂Te₃. However, employing the wind chime model yields a different conclusion when considering an ab initio calculation to investigate further interactions between NMP-2D Bi₂Te₃ and IPA-2D Bi₂Te₃ separately.

2.4.3 Ab initio study on 2D material–solvent interaction and effect on time evolution. Fig. 3a shows the evolution of the number of rings with respect to time for the IPA-2D Bi₂Te₃ system. In Fig. 3b, the calculated values of for both NMP-2D Bi₂Te₃ and IPA-2D Bi₂Te₃ systems were found to be very close, as the nonlinear coefficient value depends on the material itself. In Fig. 3c and d, the rise times for IPA-2D Bi₂Te₃ and NMP-2D Bi₂Te₃ were found to be 0.213 s and 0.1139 s. Previous investigations show lower rise times for NMP-2D Bi₂Te₃ than for IPA-2D Bi₂Te₃. In the SI Table S2, the SSPM formation times reported for the diffraction pattern are taken from the recent literature. However, the time required to obtain the diffraction pattern with the maximum number of rings is longer for NMP-2D Bi₂Te₃ than for IPA-2D Bi₂Te₃.

Fig. 3 Comparative study of time evolution in IPA and NMP samples, a first principles angle of observation: (a) time evolution of the diffraction pattern for IPA at a wavelength of 650 nm and concentration C₂. (b) for IPA and NMP at a wavelength of 650 nm and concentration C₂. (c and d) Rise times of the diffraction patterns for the IPA-2D Bi₂Te₃ and NMP-2D Bi₂Te₃ solutions. Top and side views of the molecules on top of the 2D Bi₂Te₃ surface. (e) C1-IPA, (f) C2-IPA, (g) C1-NMP and (h) C2-NMP. The pink, yellow, black, blue, red, and white spheres represent the Bi, Te, C, N, O, and H atoms, respectively. The red dashed lines denote the simulation unit cell. Side views of the differential charge density plots for (i) C1-IPA (isovalue 1.3 × 10⁻⁴ e Bohr⁻³) and (j) C1-NMP (isovalue 2.5 × 10⁻⁴ e Bohr⁻³). Regions of charge accumulation and depletion are denoted in yellow and cyan, respectively. Top views of Bader charge analysis (k) for C1-IPA and (l) for C1-NMP. Projected density of states per atom for the valence states of (m) C1-IPA and (n) C1-NMP.

We have carried out a series of computer simulations to understand the interactions between 2D Bi₂Te₃ and the molecules (IPA or NMP). The computational details are provided in the SI section S4. We have considered four configurations: two for IPA (C1-IPA and C2-IPA) and two for NMP (C1-NMP and C2-NMP), as illustrated in Fig. 3e, f and g, h. All optimized configurations exhibited negative binding energies, indicating that both IPA and NMP can potentially bind to the 2D Bi₂Te₃ surface. The binding energies were computed using eqn (12) (SI), and the obtained values were −0.30 eV, −0.29 eV, −0.56 eV, and −0.26 eV for C1-IPA, C2-IPA, C1-NMP, and C2-NMP, respectively. Since all binding energies are negative, all four configurations are viable. However, a more negative binding energy indicates a stronger interaction. Therefore, C1-NMP was identified as the most favorable configuration.

The vertical distance between the molecules and the surface ranged from 1.91 Å to 2.80 Å, with C1-NMP exhibiting the shortest distance, suggesting a stronger interaction, consistent with its most negative binding energy. Table 3 summarizes the binding energies and the shortest vertical distances between the molecules and the 2D Bi₂Te₃ surface.

Table 3 Binding energy values and shortest vertical distances for the optimized configurations

Configuration	E b (eV)	Distance (Å)
C1-IPA	−0.30	2.50
C2-IPA	−0.29	2.52
C1-NMP	−0.56	1.91
C2-NMP	−0.26	2.80

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We then analysed the interactions between C1-IPA and C1-NMP using differential charge density analysis (eqn (13) in the SI) and Bader charge analysis.⁵⁰ As depicted in Fig. 3i for C1-IPA and Fig. 3j for C1-NMP, there is charge accumulation (highlighted in yellow) between Te and the molecules, suggesting a significant interaction between them. Moreover, as illustrated in Fig. 3k for C1-IPA and Fig. 3l for C1-NMP, Bader charge analyses indicate that there are no significant changes at the surface; i.e., only a slight charge transfer occurs between the molecules and 2D Bi₂Te₃, as we mainly have van der Waals (vdW) interactions.

Furthermore, in Fig. 3m and n, we have examined the projected density of states (pDOS) of the C1-IPA and C1-NMP systems to obtain deeper insights. Fig. 3m and n display the pDOS projected onto the s states of the H atoms and the p states of the Bi, Te, C, O, and N atoms for the C1-IPA and C1-NMP systems, respectively. Both systems exhibit an electronic band gap value of approximately 1.30 eV. Additionally, the pDOS indicates a stronger interaction for C1-NMP compared to C1-IPA, as evidenced by the observed hybridization in the C1-NMP case between the more prominent peak of the p orbital of the Te atom and the p orbitals of the O and N atoms near the valence band maximum (VBM).

These results align with the experimental findings (see Table 2). The rise time (τ_rise) for NMP-2D Bi₂Te₃ is shorter than that for IPA-2D Bi₂Te₃ due to the higher polarization observed in NMP-2D Bi₂Te₃. This increased polarization enhances the likelihood of electron–hole pair generation under the same laser illumination intensity. Consequently, the torque generated under an identical electric field is more pronounced for NMP-2D Bi₂Te₃ compared to IPA-2D Bi₂Te₃. The time duration needed to attain the maximum number of rings is longer for IPA-2D Bi₂Te₃ than for NMP-2D Bi₂Te₃. The wind chime model can be used to explain this behaviour, where 2D Bi₂Te₃ and its solvent interface form a rotating domain. For NMP-2D Bi₂Te₃, this rotation takes less time because stronger molecular interactions with the solvent reduce the effective viscosity of the system. In contrast, the IPA-2D Bi₂Te₃ system exhibits weaker molecular interactions, thus allowing a longer rotation time period for a unitary domain under laser illumination of the same intensity. The conclusion drawn is as follows: the initial torque experienced by NMP-2D Bi₂Te₃ is higher than that of IPA-2D Bi₂Te₃, resulting in a shorter rise time (τ_rise) for NMP-2D Bi₂Te₃. As time progresses, the polarized NMP-2D Bi₂Te₃ structure also polarizes the surrounding NMP molecules. Consequently, it experiences a weaker drag force (fluidic viscous force) than IPA-2D Bi₂Te₃. Hence, NMP-2D Bi₂Te₃ in the cuvette solution takes a lesser time to align itself with the electric field and to exhibit a complete AC electric coherence. The intensity-dependent dynamic collapse of the diffraction pattern and the variation in the nonlinear refractive index with different solvents are discussed in the SI section S11.

The total pattern formation time for IPA is 1.2 s and for NMP is 0.9333 s. The calculated time for the maximum number of rings to appear is documented in Table 4. Two different solvents are used to realize a fast-response all-photonic isolator, although it is ultimately found that viscosity plays a more important role in pattern formation.

Table 4 Rise time, time needed for the maximum number of rings to appear, time taken for full vertical diameter rings to appear, collapse time for full vertical distortion observed in the diffraction pattern, and time required for the diffraction rings to reach a stable condition based on solvent type, wavelength, and intensity of the laser beam

Sample	Wavelength (nm)	Intensity (W cm^−2)	Rise time fitted (τrise [s])	Time needed for the maximum number of rings to appear (s)	Time taken for full vertical diameter rings to appear (s)	Collapse time for full vertical distortion (s)	Time required for the diffraction rings to reach a stable condition (s)
NMP-2D-Bi2Te3	650	10.44	0.1139	0.3333	0.3666	0.5666	0.93333
IPA-2D-Bi2Te3	650	10.44	0.213	0.5	0.5	0.7	1.2
NMP-2D-Bi2Te3	532	5.3	0.1666	0.5333	0.5333	1.2	1.7333
NMP-2D-Bi2Te3	405	1.29	0.106	0.3	0.3333	1.3	1.6333

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Total pattern formation time = time taken for full-diameter rings to appear + time required for the diffraction rings to reach a stable condition.

In other words, total pattern formation time ≈ time needed for the maximum number of rings to appear + time required for the diffraction rings to reach a stable condition, as the volume of the total 2D Bi₂Te₃ present in the solvent is small compared to the volume of the solvent. By comparing under the same wavelength (650 nm) of the laser beam at a particular intensity and constant concentration of the 2D Bi₂Te₃-NMP system, it takes the shortest time for stable diffraction pattern formation. Thus, NMP was selected as the solvent for the realization of the all-photonic isolator at three different wavelengths. The 2D structure of the nanoparticles contributes to an improved rise time compared to other microparticles like TaAs and MoP.^5,48

2.5 2D-Bi₂Te₃/2D-hBN-based nonlinear all-optical isolator

To further investigate the photonic isolators exploiting the nonlinear optical (NLO) phenomena, a unique nonlinear photonic isolator was created by utilizing a hybrid structure of 2D Bi₂Te₃ and 2D hBN.^51–53 The 2D hBN material was synthesized using the LPE process, and it exhibits a larger optical bandgap of 5.38 eV compared to 2D Bi₂Te₃. The approach used to estimate the 2D hBN optical band gap is described in section S8 of the SI. Consequently, this combination of 2D Bi₂Te₃ and 2D hBN could be used in all-photonic isolator applications.^41,43 The all-photonic isolator was realized using laser beams at 650, 532, and 405 nm, employing SSPM spectroscopy. When the forward-biased 2D-Bi₂Te₃/2D-hBN configuration is realized, as shown in Fig. 4a, a diffraction pattern is observed. This pattern can be observed in Fig. 4c(①–⑧), Fig. 4e(①–⑧), and Fig. 4g(①–⑦) for wavelengths of 650, 532, and 405 nm, respectively.

Fig. 4 Depiction of an all-photonic isolator. (a and b) Schematic showing forward- and reverse-biased analogy for the all-photonic diode. (c–h) The diffraction rings vs. intensity under forward-bias conditions and reverse-bias conditions, respectively, at different wavelengths (λ = 650 nm, 532 nm, and 405 nm). (i–k) Experimental values of obtained for asymmetric light propagation at different wavelengths (λ = 650 nm, 532 nm, and 405 nm). (l and m) Diagrams illustrating the mechanism of the all-photonic isolator based on band structure information under forward and reverse conditions.

In the second configuration illustrated in Fig. 4b, where the 2D hBN solution cuvette is placed before the 2D Bi₂Te₃ cuvette, the beam profile consistently displays a Gaussian shape. This is due to the reverse saturation property of 2D hBN, which reduces the strength of the laser beam transmitted through the cuvette. This low-intensity laser beam does not produce a diffraction pattern because it does not exceed the threshold level. Hence, the laser beam is unable to excite a diffraction pattern in the 2D Bi₂Te₃ cuvette. The 2D hBN material exhibits reverse saturable absorption, enabling the creation of an all-optical isolator.⁵⁴ The Gaussian beam profiles are shown in Fig. 4d(①–⑧), Fig. 4f(①–⑧), and Fig. 4h(①–⑤) for wavelengths of 650, 532, and 405 nm, respectively. The calculated values of for the different wavelengths (650, 532, and 405 nm) are 1.14, 4.34, and 7.46 cm² W⁻¹, respectively, under forward bias. The observed results are similar to those obtained from a single cuvette containing a 2D Bi₂Te₃-NMP solution. Fig. 4i–k displays the linear regression graphs for the forward and reverse bias conditions. Asymmetric light propagation is achieved by employing three distinct laser types, each with a different wavelength of λ = 650, 532, and 405 nm. Each of these wavelengths corresponds to a photon energy exceeding the bandgap of 2D Bi₂Te₃, which is 0.9 eV. Thus, the laser photons can induce band-to-band transitions. In the forward configuration illustrated in Fig. 4l, photons from the incoming laser beam excite valence-band electrons, prompting their transition to the conduction band. Subsequently, the electrons that have gained energy decay to their original states, emitting photons in the process. The photons, with distinct phases, interact with the laser beam, resulting in a diffraction pattern.^14,41 The electrons in the conduction band oscillate in a direction opposite to the electric field of the laser beam, resulting in the generation of charges with opposite polarities in the suspended material.¹¹ Upon interaction with the incoming laser beam, the polarized flakes will align themselves with the electric field axis of the laser beam in order to minimize their interaction energy. The nonlinear optical response of 2D Bi₂Te₃ is enhanced, leading to the observation of the optical Kerr effect.^14,40 The band-to-band transitions in 2D hexagonal boron nitride (2D hBN) cannot be induced by laser beams with wavelengths of λ = 650, 532, and 405 nm. Electrons dissipate energy during an intraband transition. The reverse saturable absorption feature of 2D hBN causes a decrease in the incoming beam intensity below a certain threshold. This prevents the generation of a diffraction pattern from the 2D Bi₂Te₃-NMP solution, as shown in Fig. 4m. The claimed all-optical isolator operates over a range of applicable wavelengths. A similar technique may be used to determine the value of n₂ for the 2D Bi₂Te₃-based photonic diode, as explained in the SI section S10. In the SI section S6, the relationship between χ⁽³⁾_monolayer on m* and χ⁽³⁾_monolayer on µ is obtained, and the identification between N-type and P-type Bi₂Te₃ is explored.

3. Conclusion

This work demonstrates the synthesis of high-quality 2D Bi₂Te₃via liquid-phase exfoliation. We have determined the mechanisms and underlying physical principles of SSPM pattern formation using 2D Bi₂Te₃. We have examined the observed phenomena using the windchime model while considering the electronic coherence theorem. The nonlinear optical coefficients, such as n₂ and χ⁽³⁾_total, were computed. The values of n₂ were found to be 2.7 × 10⁻⁴, 8.14 × 10⁻⁴, and 10.1 × 10⁻⁴ cm² W⁻¹ at the specified wavelengths of 650, 532, and 405 nm. The time required for diffraction pattern formation was measured at different wavelengths (650, 532, and 405 nm) while keeping other variables constant. In this study, the computed values of the third-order nonlinear susceptibility for a monolayer (χ⁽³⁾_monolayer) are 1.2 × 10⁻⁷ e.s.u., 3.63 × 10⁻⁷ e.s.u., and 4.51 × 10⁻⁷ e.s.u. for the wavelengths 650, 532, and 405 nm. The calculated values of χ⁽³⁾_total were comparable to those of other 2D materials. The time evolution of the diffraction patterns for different wavelengths (650, 532, and 405 nm) and solvents were also calculated. The study of diffraction ring formation under SSPM, using two solvents, demonstrated that the rise time (τ_rise) increased with increasing solvent viscosity, consistent with the wind chime model. First-principles calculations using Bader charge analysis were performed to examine the interaction mechanisms between the solvent molecules (NMP and IPA) and 2D Bi₂Te₃. NMP shows stronger interaction with 2D Bi₂Te₃, which could be related to the decrease in the effective viscosity. On the other hand, IPA exhibited weaker molecular interactions, leading to slower rotation under the same applied electric field. The distortion in the vertical direction of the SSPM pattern was quantified for each wavelength (650, 532, and 405 nm), and the collapse time for both solvents (NMP and IPA) was evaluated under 650 nm wavelength at 10.44 W cm⁻² laser illumination. The derived experimental conclusions were used to fabricate a solution–2D nanostructure system capable of exhibiting a fast response to laser beam illumination while maintaining a wide wavelength response. A fast-operating novel configuration of a nonlinear photonic isolator has been designed to facilitate nonreciprocal light propagation. This isolator comprises a 2D-Bi₂Te₃–2D-hBN heterostructure. The all-photonic isolator demonstration in this work shows values of 1.14, 4.34, and 7.46 W cm⁻² for wavelengths of 650, 532, and 405 nm under forward-bias conditions. We have characterized the photonic diodes based on the reported dN/dI value, χ⁽³⁾_total, n₂, and the operating range based on intensity and included the results in the SI Table S9. To further differentiate this effect from the thermal lens effect, the dependence of χ⁽³⁾_monolayer on m* and χ⁽³⁾_monolayer on µ was determined, resembling the pertinent curves documented by other authors. The high values of n₂ and χ⁽³⁾_total are attributed to laser-induced hole coherence.

Author contributions

Saswata Goswami: conceptualization, methodology, software, formal analysis, investigation, data curation, writing – original draft, and visualization. Bruno Ipaves: conceptualization, methodology, software, formal analysis, investigation, data curation, writing – original draft, and visualization. Juan Gomez Quispe: methodology, software, formal analysis, data curation, and visualization. Caique Campos de Oliveira: methodology, software, formal analysis, data curation, and visualization. Surbhi Slathia: formal analysis, investigation, and data curation. Abhijith M. B.: investigation and data curation. Varinder Pal: methodology, formal analysis, investigation, and data curation. Christiano J. S. de Matos: investigation and writing – review & editing. Samit K. Ray: methodology, writing – review & editing, and supervision. Douglas S. Galvao: investigation and writing – review & editing. Pedro A. S. Autreto: conceptualization, methodology, validation, investigation, resources, writing – review & editing, supervision, project administration, and funding acquisition. Chandra Sekhar Tiwary: conceptualization, methodology, validation, investigation, resources, writing – review & editing, supervision, project administration, and funding acquisition.

Conflicts of interest

There are no conflicts to declare.

Data availability

The data supporting this article have been included as part of the supplementary information (SI). Supplementary information is available. The 2D-Bi₂Te₃ synthesis, characterizations (XRD, TEM, AFM, SEM, UV-Vis, XPS, Raman Spectroscopy) are available in the Supporting Information. Theoretical estimation of n₂ and χ(3)total, electronic relationship between χ(3)monolayer on m* and χ(3)monolayer on µ, intensity-dependent dynamic collapse of diffraction pattern, and variation in nonlinear refractive index with different solvents, and intensity-dependent temporal evolution of diffraction pattern for different solvents are included in the SI. See DOI: https://doi.org/10.1039/d6nr00354k.

Acknowledgements

C. S. T. acknowledges the Core research grant of SERB, India; STARS projects by MHRD, India; the DAE Young Scientist Research Award (DAEYSRA); the AOARD (Asian Office of Aerospace Research and Development) grant no. FA2386-21-1-4014; and the Naval Research Board for funding support. D. S. Galvao acknowledges financial support from FAPESP/CEPID Grant 2013/08293-7, INEO/CNPq and FAPESP grant 2025/27044-5. B. I. acknowledges CNPq process numbers #194155/2025-0 and FAPESP process numbers #2024/11016-0. C. C. O. acknowledges the Sao Paulo Research Fundation (FAPESP process number 2024/11376-6). P. A. S. A. acknowledges Coordenaco de Aperfeicoamento de Pessoal de Nivel Superior (CAPES finance code 001), CNPq (grant 308428/2022-6) and National Institute of Science and Technology on Materials Informatics (grant no. 371610/2023-0). C. C. O., B. I., J. G. Q. and P. A. S. A. also acknowledge the UFABC Computation Multiuser Center (CCM) for the computational resources provided. Team acknowdgles SPARC (The Scheme for Promotion of Academic and Research Collaboration) by ministry of education, India.

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Footnote

† These authors contributed equally.

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