## Abstract

The nonlinear optical properties of Bi_{2}Se_{3} nanosheets of different diameters were studied using the femtosecond Z-scan technique. As the excitation wavelength increased from that for resonant absorption to non-resonant absorption, the nonlinear absorption of the smaller nanosheets with diameter of 30 or 80 nm changed from saturation to reverse saturation and the corresponding relaxation times decreased, unlike for larger nanosheets with diameter of 2000nm. The more sensitive nonlinear refractive index changed from positive to negative for all the nanosheets, when the excitation wavelengths were near-resonance absorption wavelengths. A simplified model similar to noble metal nanomaterials explains the observations suitably. The reversal in optical nonlinearities of Bi_{2}Se_{3} nanosheets vastly enhances their properties.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Bi_{2}Se_{3} has a topologically nontrivial energy gap of 0.3 eV [1,2], that make it a potential broadband saturable absorber. When it is exfoliated to few-layer sheet, Bi_{2}Se_{3} nanosheets have been confirmed as the excellent saturated absorbers in the visible to mid-infrared band [3] without reverse saturated absorption (RSA). And its nonlinear refractive properties were reported as self-focusing effects at 800 nm, where nonlinear refractive index ${n_2} > 0$ [4,5].

The characteristic of topological insulator is the metal layer on the surface. For Bi_{2}Se_{3} films, it was reported that the thickness of metal phase on the surface is between 6 nm and 12 nm [6]. The metal layer on the surface of a topological insulator is similar to that of noble metals [3]. It is known that surface plasmon resonance (SPR) in noble metal nanostructures can modulate the linear absorption [7], nonlinear absorption, and refractive properties [8,9] by changing the structure limit parameters, which is potentially useful for developing new photonic devices [10]. Different from the films, the limited hexagonal shape of Bi_{2}Se_{3} nanosheets may arouse the similar SPR effect, leading to some significant result such as reverse saturated absorption or self-defocusing effects.

Very recently, SPR was introduced in 2D nanomaterials. For example, in 2018, Wang et al. have prepared molybdenum oxide nanosheets, which have a metallic character and SPR peak in the visible region. It suggests that plasmon resonance can enhance the nonlinear optical absorption [8]. This is because, during photobleaching recovery, both the fast and slow relaxation processes in the plasmon resonance region are longer than those in the non-resonance region, indicating that the plasma has a stronger response to nonlinear optical absorption [8]. A sign reversal in nonlinear refraction has been reported in glass-metal nanocomposites when the interaction of light is switched from near-resonance to non-resonance [9] signifying that nonlinear refraction can also be modulated by SPR. If the similar effects of reversal in optical nonlinearities could be observed and modulated in some 2D nano-materials [11–15], it will be very useful for potential applications in this field. But the controllable shapes and sizes are the key for the controllable plasmon resonance, so Bi_{2}Se_{3} nanosheets with hexagonal shapes seems an ideal research target.

As the surface of a topological insulator is metallic, a change in diameter of Bi_{2}Se_{3} nanosheets in the scale of 30-2000 nm affects the position of the absorption theoretically, which in turn affects their nonlinear optical properties at different excitation wavelengths. In this study, by changing the excitation wavelengths close to or far from the absorption peaks of the Bi_{2}Se_{3} nanosheets, we observed clear sign changes from saturated absorption to reverse saturated absorption in the nonlinear absorption and self-focusing to self-defocusing in the nonlinear refractive properties, contrary to previous reports on broadband saturation absorption and self-focusing. These significant results are attributed to surface-assisted two-photon absorption and negative values of the local field factor *f*.

## 2. Experiment and discussion

Three types of Bi_{2}Se_{3} nanosheets with hexagonal shape were prepared according to the methods reported previously [16,17]. The TEM and AFM results of previously published papers indicated that small-sized nanosheets, of diameters 30 nm and 80 nm and thicknesses 2 nm and 8 nm, respectively, are approximately hexagonal in shape [16], whereas the shape of larger Bi_{2}Se_{3} nanosheets of diameter 2000nm and thickness 30 nm is strictly hexagonal [17]. The corresponding models and TEM are given in the insets of Fig. 1.

The absorption spectra of the three Bi_{2}Se_{3} nanosheets are presented in Fig. 1. The morphologies of Bi_{2}Se_{3} nanosheets were characterized by TEM, and the average sizes of three Bi_{2}Se_{3} nanosheets are determined to be approximately 30 nm, 80 nm, and 2000nm. Figs. 1(a) and 1(b) show that the smaller nanosheets have larger absorption at shorter wavelengths with an inconspicuous absorption peak between 400 and 600 nm. The finite-difference time-domain (FDTD) method was used to simulate the absorption spectra. The hexagonal prism models of diameter 30 nm and thickness 2 nm and diameter 80 nm and thickness 8 nm are shown as insets in Figs. 1(a) and 1(b), respectively. As the thickness of the metal layer was reported to be approximately (6 + 6) 12 nm in the film [6], the metallic nature of these nanosheets was considered to be dominant. The values of the refractive index were set similar to Au in the FDTD programs. The yellow dotted lines represent the results of the FDTD simulation and indicate that the absorption is larger at shorter wavelengths and weaker at longer wavelengths.

For the larger nanosheets, the absorption peak is clearly seen at 660 nm in Fig. 1(c). The hexagonal prism model of diameter 2000nm and thickness 30 nm is shown as the inset in Fig. 1(c). However, the refractive indexes were set as followed: an insulating layer of 18 nm were sandwiched between two metal layers of 6 nm, whose refractive index is similar to that of the bulk Bi_{2}Se_{3}. The yellow dotted line shows the simulation result in Fig. 1(c) with a clear visible absorption peak.

To study the influence of metal layer thickness or diameter on the absorption spectra of nanosheets, the contrastive simulation results are presented in Fig. 2. The thickness of the metal layers was halved (6 nm to 3 nm) and the result is shown as the blue dotted line in Fig. 2, which has a shape similar to that in a previous calculation. The diameter of the nanosheets was halved (2000 nm to 1000 nm) and the result shown as the red dotted line in Fig. 2 indicates that the shape of the absorption peak is distinctly different.

The simulation results in Fig. 2 reveal a possibility that the absorption peak of the larger nanosheets in Fig. 1(c) is mainly influenced by the restricted diameter, similar to the size effect in Au, Ag nanomaterials [18]. The size limitation in Au, Ag nanomaterials leads to visible surface plasma absorption peaks, which can modulate the nonlinear absorption properties [19]. Therefore, open-aperture (OA) and closed-aperture (CA) Z-scan tests were performed on the three Bi_{2}Se_{3} nanosheets, using excitation wavelengths of 400 nm, 660 nm, and 800 nm, to study the effects of resonance and non-resonance absorption on nonlinear absorption and refraction [20].

In OA and CA Z-scan experiments, a femtosecond pulse laser system of pulse duration 35 fs and repetition rate 2 kHz was used. The fs pulse width and the interval as 0.5 millisecond(ms) ensured that the thermal effect, if existed, could not be accumulated. A lens of 150 mm focal length focused the laser beam and a detector (OPHIR, PD300R-IR) received the transmitted laser beam. The Z-scan system was calibrated using a CdS film as a standard sample before the experiment. It was ascertained that there was no signal for solvents under the same conditions. The CA results were obtained from the original CA/OA Z-scan data, excluding the influence of the OA data. The CA Z-scan system was calibrated using a CS_{2} solution as a standard liquid sample. The results of CS_{2} shown the self-focusing characteristic, meaning little thermal effect existed.

The OA Z-scan results of the smaller nanosheets are shown in Figs. 3(a) and 3(f). Their nonlinear absorption characteristics are similar, owing to their similar linear absorptions. When the excitation wavelength of 400 nm is close to the absorption peak, it exhibits saturated absorption as shown in Figs. 3(a) and 3(d). When the excitation wavelength increases to 660 nm, slightly away from the absorption peak, the saturated absorption becomes weaker as shown in Figs. 3(b) and 3(e). When the excitation wavelength increases to 800 nm, the reverse saturated absorption can be observed, as shown in Figs. 3(c) and 3(f). With the increase of excitation intensity at 800 nm, the valley deepens noticeably, signifying that the dominant mechanism is two-photon absorption, that is rarely reported in Bi_{2}Se_{3} nanoplatelets or films [3,21]. The nonlinear optical scattering experiments of Bi_{2}Se_{3} nanosheets with R = 30 nm and R = 80 nm were carried out at the wavelength of 800 nm. The results show the scattering light intensities are linear dependent on input laser intensities and the nonlinear optical scattering had little influence. In the case of the larger nanosheets, saturated absorption is always dominant in the visible wavelength region, whether the excitation wavelength is slightly shorter or longer than the resonance absorption peak. Even under resonance absorption, the weak saturated absorption remains unchanged, which may be due to the increased contribution of the bulk state to the nonlinear response [3]. In summary, Bi_{2}Se_{3} nanosheets of diameters 2000nm maintain a constant saturation absorption peak in the visible band, which makes it a well-saturated absorber in the broadband. However, the rare reversal of nonlinear absorption in the smaller nanosheets from saturated absorption to reverse saturated absorption could greatly augment the applicability of Bi_{2}Se_{3} nanosheets at excitation wavelengths far removed from the resonance peaks.

Monochromatic pump-probe tests were carried out at 400 nm and 800 nm on the three nanosheets. The pump-probe system could detect only the absolute value of the transient transmittance change, because that a chopper and the corresponding lock-in amplifier were used for improved sensitivity. The lock-in amplifier only gives the absolute value of the change, not including negative and positive signals. The results are shown in Fig. 4, which is fitted by a single exponential decay function. The relaxation times of all the nanosheets are in the picosecond scale, indicating ultrafast dynamic processes. For the larger nanosheets of diameters 2000nm, the lifetimes at 400 and 800 nm are approximately 2.2 ps and 1.9 ps, respectively, which are almost the same. The corresponding mechanism is that photoexcited hot carriers are generated by electron-longitudinal optical-phonon scattering, as in thick films [6].

The lifetimes for the nanosheets of diameters 30 nm and 80 nm are 58 ps and 48 ps, respectively, at 400 nm. There are two possible explanations for the corresponding relaxation mechanism. One is the interband relaxation involving the Dirac 2D surface state [22] and another is the prolonged internal thermalization of a nonequilibrium electron population due to electron-lattice interactions, as in noble metal thin films [6]. When the excitation wavelength is 800 nm, the lifetimes of the nanosheets reduced to 33.3 ps and 11.7 ps; the relaxation time decreases as the wavelength is far from that of resonance absorption and the corresponding prolongation weakens. The bi-chromatic or white-light pump-probe spectra tests will be carried out for the in-depth mechanism study in the future.

The results of the CA Z-scan obtained from the original CA Z-scan data/OA Z-scan data are shown in Fig. 5. For the smaller nanosheets, the pre-focal peak and post-focal valley at 400 nm (near the resonance absorption) suggest a negative sign of the complex nonlinear refractive index, ${n_2} < 0$, which has not been reported. As the excitation wavelengths move away from resonance to 660 nm and 800 nm, both the results show a pre-focal valley and post-focal peak, ${n_2} > 0$, which is the same as previously reported [4].

For the larger nanosheets, the same inversion occurs at the resonant wavelength of 660 nm, exhibiting a pre-focal peak and the post-focal valley, ${n_2} < 0$. As the excitation wavelengths move away from resonance to 400 nm or 800 nm, both the results show a pre-focal valley and post-focal peak, ${n_2} > 0$. All the nanosheets have ${n_2} < 0$ near the resonance absorptions; this could be caused by an increase in the local field factor *f* in the resonance region, which involves the hot electron contribution, as in noble metal nanomaterials [9]. Detailed explanations and simplified models will be given later.

To sum up: when the excitation wavelengths were near the resonance absorptions, the nonlinear refractive properties, which are more sensitive than nonlinear absorption, reversed from ${n_2} > 0$ to ${n_2} < 0$ for all the samples. This phenomenon is a noticeable and valuable feature because different phase distributions or focus states can be obtained from opposite signs of ${n_2}$ when light of different wavelengths passes through the same Bi_{2}Se_{3} nanosheets. That is very useful for the wavelength-dependent dispersion device and others, if the resonant wavelengths of Bi_{2}Se_{3} nanosheets could be modulated to some special bands by the shapes and sizes.

The fitted nonlinear optical parameters of the three Bi_{2}Se_{3} nanosheets at different wavelengths are shown in Table. 1. The results in blue have not been reported, but the results in red can be compared with those reported earlier. The fitting results of the Bi_{2}Se_{3} nanosheets of diameter 2000nm at 800 nm are within the same order of magnitude as those reported by Lu [4] or Wang [3].

The nonlinear parameters thus obtained were converted for further analysis into real and imaginary parts of ${\chi ^{(3 )}}$, which represent the nonlinear optical properties of Bi_{2}Se_{3}. Figure 6 shows the results as a function of the difference between the resonance absorptions and excitation wavelengths. As the resonance absorptions do not have a definite value, but extend over an approximate range, the distribution of the nanosheets in the dispersion is incidental, the shape of the nanosheets is not uniform, and is anisotropic. Hence, different samples cannot be compared quantitatively. However, the results are similar to those of noble metal nanomaterials [9]: the imaginary part is symmetric with resonance (${\Delta \lambda } = 0$) whereas the real part shows an evident sign change. The two spots shown in the top left corner of Figs. 6(a) and 6(b) are the exceptions. These two positive values at ${\lambda } = 800\,\textrm{nm}$ correspond to two-photon absorption and are consistent with previous results [9].

A corresponding analysis based on a simplified model similar to noble metal nanomaterials is presented. For femtosecond pulses, the main contribution of Bi_{2}Se_{3} nanosheets to ${\chi ^{(3 )}}$ is considered to be the hot electron contribution. Similar to the noble metal nanomaterials in a previous work [9], the spectral and intensity dependence of this contribution or total susceptibility can be modeled in the simplest approximation by a Lorentzian-shaped ${\chi ^{(1 )}}$, such as $\langle{{\chi }(I )} \rangle = {\chi ^{(1 )}} \cdot {[{1 + ({I/{I_S}} )} ]^{ - 1}}$, ${\Delta \chi } = \langle\chi (I )\rangle- {\chi ^{(1 )}}$, and ${\Delta \chi } = 3{\pi }{\chi ^{(3 )}}{|{{E_1}} |^2}$, where ${I_S}$ is the saturation intensity. Therefore, the real and imaginary part of ${\chi ^{(3 )}}$ should have an approximate Lorentz shape, influenced by linear susceptibility, as a function of the difference between the resonance absorptions and excitation wavelengths.

Further, the local field factor *f* can be calculated based on the relation ${E_1} = f \cdot {E_0}$ from Fig. 6 (b). Based on the above analysis and an effective medium theory, the real part of ${\chi ^{(3 )}}$ is attributed to the hot electron contribution of the Bi_{2}Se_{3} surface metal layer conduction band electrons, similar to noble metal nanomaterials [9]. The metal volume fraction $p$ can be used to describe the nonlinear local field $\chi _{eff}^{(3 )} = p{f^2}{|f |^2}\chi _m^{(3 )}$. Therefore, if we set $\textrm{Re}{\chi ^{(3 )}}({{\Delta \lambda } = 320\textrm{nm}} )$ of $f \approx 1$, we can obtain the local field factor. For example, $\textrm{Re}{\chi ^{(3 )}}({{\Delta \lambda } = 60\textrm{nm}} )\approx 16 \cdot \textrm{Re}{\chi ^{(3 )}}({\Delta \lambda = 320\textrm{nm}} );$ therefore, $|f |\approx 2$ at $\textrm{Re}{\chi ^{(3 )}}({{\Delta \lambda } = 60 \textrm{nm}} )$ can be obtained. Thus, an effective medium approach, such as in noble metal nanomaterials, is used to describe the third-order optical nonlinearity of topological insulator nanosheets.

## 3. Conclusion

Using the femtosecond Z-scan technique, we established the sign changes in the nonlinear optical properties in Bi_{2}Se_{3} nanosheet dispersions with diameters of 30, 80, and 2000 nm, which are affected by resonance absorptions. For nonlinear absorptions, in the smaller nanosheets, the saturated absorption reversed to reverse saturated absorption when the excitation wavelengths increased from 400 nm for resonance absorption to 800 nm for non-resonance absorption, whereas the corresponding relaxation times reduced. This was attributed to the dominant mechanism of two-photon absorption at the non-resonance wavelengths. By contrast, the nonlinear absorptions and relaxation times of the larger nanosheets remained unchanged as the corresponding mechanism of generation of photoexcited hot carriers occurred by electron-LO-phonon scattering. In the case of nonlinear refraction, when the excitation wavelengths were near the resonance absorption, the nonlinear refractive indexes changed from positive to negative in all the nanosheets. This can be explained by the electric field enhancement due to surface plasmon resonance in the Bi_{2}Se_{3} surface metal layer, similar to noble metal nanomaterials. The results of positive nonlinear absorption index and negative nonlinear refractive index of Bi_{2}Se_{3} nanosheets can greatly enhance its applicability in optical communication.

## Funding

National Natural Science Foundation of China (NSFC) (61874141, 61875232); Fundamental Research Funds for the Central Universities of Central South University (2017zzts327).

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