## Abstract

The third order nonlinear optical property of Bi_{2}Se_{3}, a kind of topological insulator (TI), has been investigated under femto-second laser excitation. The open and closed aperture *Z*-scan measurements were used to unambiguously distinguish the real and imaginary part of the third order optical nonlinearity of the TI. When excited at 800 nm, the TI exhibits saturable absorption with a saturation intensity of 10.12 GW/cm^{2} and a modulation depth of 61.2%, and a giant nonlinear refractive index of 10^{−14} m^{2}/W, almost six orders of magnitude larger than that of bulk dielectrics. This finding suggests that the TI: Bi_{2}Se_{3} is indeed a promising nonlinear optical material and thus can find potential applications from passive laser mode locker to optical Kerr effect based photonic devices.

©2013 Optical Society of America

## 1. Introduction

Optical materials with a remarkably large third-order nonlinear susceptibility *χ*^{(3)}, or a large *nonlinear refractive index n*_{2}, are highly in demand for all-optical signal processing applications, e.g. in ultra-short optical pulse shaper, optical switcher, wavelength converter, *etc* [1]. In the recent years, many materials had been experimentally confirmed to show large nonlinear refractive index. Conventional bulk semiconductors like GaAs and silicon possess a *nonlinear refractive index n*_{2} = 1.59 × 10^{−17} m^{2}/W and *n*_{2} = 0.45 × 10^{−17} m^{2}/W at telecom wavelengths, respectively [2]. For example, various quantum dots (QDs) studied at different wavelength: ZnO QDs at 1064 nm [3], ZnS QDs and Mn^{2+} doped ZnS QDs at 532nm wavelength [4]. In addition to these zero dimensional nano-materials, other 2-dimensional layer-by-layer new materials, such as graphene, were also experimentally found to show many interesting nonlinear optics properties. Unlike conventional bulk materials, the electrical and optical properties of few-layer materials can be effectively engineered over a wide spectral range by chemical or electrical doping, offering a great flexibility in tailoring their nonlinear optical characteristics. In addition, two-dimensional materials with layer geometry have also the intrinsic advantage of compatibility with the mature CMOS (complementary metal oxide semiconductor) technology. The imaginary part of the complex susceptibility in graphene, that is, saturable absorption term, has been widely investigated, and this property has led to the success in passive mode locking or Q-switching in lasers from visible band to microwave frequency band [5–15]. In the real part, large nonlinear refractive index *n*_{2} = 10^{−11} m^{2}/W was experimentally measured by Zhang *et al.* [16], making graphene promising for a number of photonic and optoelectronic applications [17].

In the parallel, the search for other new layered materials that have a comparable *nonlinear refractive index* is highly motivated. Another type of layer-by-layer material, the topological insulator (TI), has received great attentions in condensed-matter physics [18, 19]. TI is characterized by its bulk insulating state with a small band gap and a surface Dirac-like band structure, which is similar to that in graphene. Very recently, Bernard *et al.* experimentally investigated the saturable absorption of topological insulator at the telecommunication [20] and Zhao *et al.* successfully used topological insulator as an effective saturable absorber for the passive mode locking of lasers [21, 22]. Enlightened by the similarity of electronic properties and saturable absorption between graphene and TI, one may wonder whether TI also has a large *nonlinear refractive index*.

In this paper, we present results of study on the *nonlinear refractive index* of a TI: Bi_{2}Se_{3}. Under femto-second laser illumination at the near-infrared wavelength region (800 nm), we measured the real and imaginary part of the complex *nonlinear refractive index* (saturable absorption and optical Kerr nonlinearity) with the laser *Z*-scan technique. At high optical intensity a clear saturation of the change in *n*_{2} is observed. This may indicate large potentials of TI based nonlinear photonics devices for next-generation all-optical signal processing.

## 2. Experimental

#### 2.1. Characterization of TI sample

By employing the polyol method reported in Ref [23], the Bi_{2}Se_{3} nano-platelets (NPs) were synthesized, washed and dispersed in isopropyl alcohol (IPA) and dropped cast onto a common quartz plate (1 mm thick). After that, the quartz plate was put into a drying oven for evaporation over 8 hours. Figure 1(a)
shows the Raman spectra of the as-produced Bi_{2}Se_{3} sample, in which four typical Raman peaks attributed to different resonant modes could be clearly seen at low wave number range. Atomic force microscopy (AFM) was used to determine the sample thickness, as shown in Fig. 1(b). By scanning the height difference between the quartz substrate and the sample surface, the sample thickness is measured to be an average of 50 nm.

#### 2.2. Experimental setup

The *Z*-scan technique was applied to study the nonlinear optics coefficients of the as-prepared TI: Bi_{2}Se_{3}. The experimental setup is shown in Fig. 2
. The sample is subjected to femto-second pulses from a Coherent femto-second laser (center wavelength: 800 nm, pulse duration: 100 fs, 3 dB spectral width: 15 nm and repetition rate: 1 kHz). By using optical attenuators, the average power could be deliberately controlled below 1 mW, in order to ensure that the incident laser is adjusted below the optical damage threshold and the multiple-photon effect is significantly suppressed. The laser beam is then focused by an objective lens (focus length: 500 mm), generating a beam waist of 35 *μ*m, corresponding to a peak intensity up to 260 GW/cm^{2}. The TI: Bi_{2}Se_{3} sample was perpendicularly oriented towards the beam axis and translated along the *Z*-axis with a linear motorized stage. A computer controlled dual-detector power meters were used to simultaneously measure the optical power.

Measurements were performed simultaneously in two parts, an open-aperture part wherein all light transmitted through the sample is collected by the power-detector and a closed-aperture part wherein a small aperture is added before the power-detector so that only part of the on-axis transmitted beam is collected. The open-aperture measurement enables one to investigate the nonlinear absorption, while within the closed aperture measurement the change of optical transmittance is a combined consequence of the nonlinear absorption and the nonlinear phase effect induced by the optical Kerr nonlinearity. The division of the closed-aperture measurement by the open-aperture measurement results in the separation of these two effects [24]. To more precisely identify the *nonlinear refractive index*, CS_{2} solution contained in a cuvette (1 mm in thick) was used as a benchmark for the calibration (see Appendix for details). Analyzing the measured *Z*-scan data, the third-order *nonlinear refractive index* of CS_{2} is measured to be *n*_{2} = 2.66 × 10^{−19} m^{2}/W, very close to the standard value of (3 ± 0.6) × 10^{−19} m^{2}/W [25]. This result demonstrates the reliability of our *Z*-scan measurement platform.

## 3. Results and discussion

By replacing the standard CS_{2} benchmark with the TI: Bi_{2}Se_{3} sample while the other parameters were kept unchanged, *Z*-scan measurements on TI: Bi_{2}Se_{3} were performed at an incident irradiance of 10.4 GW/cm^{2}, as shown in Fig. 3
. A typical open aperture trace, when the sample is translated through the beam focus, is shown in Fig. 3(a). A sharp and narrow peak located at the beam focus clearly shows the characteristic of nonlinear absorption. A typical closed-aperture measurement was shown in Fig. 3(b). In this trace, as the effect of the nonlinear phase is of the same order of magnitude as the effect of saturable absorption, upon dividing the curve in Fig. 3(b) by the curve in Fig. 3(a), we could unambiguously verify the nonlinear phase change, as shown in Fig. 3(c). The latter has the typical shape of a *Z*-scan trace for Kerr nonlinearity. The pre-focal valley and the post-focal peak suggest the positive sign of the complex *nonlinear refractive index*, indicating the self-focusing effect in TI. Fitting the trace by the well-established formula [26],

*T*(

*x*) is the normalized transmittance,$x=z/{z}_{R}$,${z}_{R}\text{=}\pi {\omega}_{\text{0}}^{\text{2}}\text{/}\lambda $, and ΔΦ =

*kn*

_{2}

*I*

_{0}

*L*is the on-axis nonlinear phase shift at the focus,

_{eff}*k*is the wavelength number,

*I*

_{0}is the irradiance at the focus,

*L*is the sample’s effective length. The inferred nonlinear phase change ΔΦ is about 1.1 rad and the real part of the complex

_{eff}*nonlinear refractive index*is calculated to be

*n*

_{2}= 2.26 × 10

^{−14}m

^{2}/W.

*Z*-scan measurements under variable optical powers were also under investigation. Open-aperture measurement result shows that the normalized transmittance has a power dependent characteristic of saturable absorption. We fitted the curve in Fig. 4(a)
by

*T*is the transmittance,

_{${\alpha}_{s}$}is the saturable loss and

_{${\alpha}_{ns}$}is the non-saturable loss,

*I*is the input intensity and

*I*is the saturation intensity. The fitted values of

_{sat}_{${\alpha}_{s}$}and

*I*are 61.2% and 10.12 GW/cm

_{sat}^{2}, respectively. From the value of

*I*, we can calculate the nonlinear absorption coefficient that appears in Ref [27]. The high normalized modulation depth indicates that Bi

_{sat}_{2}Se

_{3}can be a suitable passive laser mode locker for the generation of ultra-short pulse.

Figure 4(b) shows the nonlinear phase $\Delta \Phi $ under different input powers. The Kerr refractive index *n*_{2} can be deduced from the slope of this curve at low intensities, based on *n*_{2} = ΔΦ / (*k*_{0}*LI*), where *k*_{0} = 2*π / λ* and *L* is the sample thickness, assumed equal to 50 nm. A value of *n*_{2} ≃ 10^{–14 }m^{2}/W is obtained, which is approximately 10^{6} times larger than that of bulk dielectrics. With the laser intensity increasing, the nonlinear phase shift decreases and saturates starting from *I* > *I _{sat}* ≈50 GW/cm

^{2}. In this high intensity regime, a more accurate modeling of the nonlinear response taking into account

*χ*

^{(3)}and higher order terms, such as

*χ*

^{(5)}. is required. However, from the experimental measurement, an effective nonlinear Kerr index could be defined by ${n}_{2}^{*}$ = ΔΦ / (

*k*

_{0}

*LI*), as a combined effect of

*nonlinear refractive index*from different order terms. One sees that ${n}_{2}^{*}$ decreases with increasing

*I*and then reaches at a constant value ${n}_{2}^{*}$ = 10

^{−14}m

^{2}/W for

*I*>50 GW/cm

_{sat}^{2}. Table1 shows the nonlinear refractive index of different optical materials and one could see that TI (Bi

_{2}Se

_{3}) has relatively large nonlinear refractive index, while TI (Bi

_{2}Se

_{3}) possesses the advantage of the two-dimensional geometry.

The electronic property of TI is characterized by the bulk insulating band and surface Dirac-cone band [28–30]. In view of that the photon energy (1.55 eV) at 800 nm is larger than the bulk insulating band gap (0.3 eV) [29], the inter-band transitions (from the cone-like valence band into the conduction band) could occur. This is because that TI has symmetric energy band structure in its band state, the momentum conservation, as a result of electron excited from the conduction band to the valance band, could be automatically fulfilled, leading to a very broad spectral range of optical absorption. In principle, TI possesses broadband saturable absorption once the photon energy is larger than its bulk band gap, and operates as another broadband optical saturable absorber, like its counterpart: graphene.

Furthermore, there are two main physical mechanisms responsible for the origin of nonlinear refraction: the bound-electronic and free-carrier nonlinearities [31]. The former effect is governed by *n*(*I*) = *n*_{0}(*I*) + Δ*n*(*I*) = *n*_{0}(*I*) + *n*_{2}*I*, while the latter follows Δ*n*(*t*) = *n*_{2}*I*(*t*) + *σ*_{γ}*N*(*t*), where *σ*_{γ} is the free carrier refraction coefficient, and *N*(*t*) is the photo-excited carrier density. We expect that these two mechanisms co-exist in the topological insulator material, and therefore, obviously, the *effective nonlinear Kerr index ${n}_{2}^{*}$* *= n*_{2} + *σ*_{γ}*N*(*t*)/*I* becomes an intensity dependent parameter, which can explain why ${n}_{2}^{*}$ changes with the incident laser intensity in the experiments, as shown in Fig. 4(b). We correlate the origin of the large Kerr nonlinearity with the Dirac Fermions in the TI: Bi_{2}Se_{3}. At low power regime, the bound-electronic nonlinearity coexists with the free-carrier nonlinearity, as a result of a large amount of bound-electrons at the bulk conduction band and free-carriers at the surface metallic band; however with the increasing of the incident light, the depletion of electrons at the conduction band induced by the saturable absorption effect weakens the contribution from the bound-electronic nonlinearity, leading to the saturation of optical nonlinearity at the high power regime, as shown in Fig. 4(b). Consequently, if one reasonably assumes that only the free-carrier nonlinearity takes effect at the high power regime (*I _{sat}* >50 GW/cm

^{2}), one can argue that

*effective nonlinear Kerr index*from the free-carrier (resp. bound-electronic) nonlinearity is about to 0.8 × 10

^{−14}m

^{2}/W (resp. 1.4 × 10

^{−14}m

^{2}/W).

## 4. Conclusion

In conclusion, the third order nonlinear optics property in TI: Bi_{2}Se_{3} was experimentally investigated by the open and closed aperture *Z*-scan technique, and its real and imaginary part of *nonlinear refractive index* (Kerr nonlinearity and saturable absorption) are separately determined, for the first time. Our measurements on the nonlinear phase and absorption yield a nonlinear coefficient *n*_{2} of 10^{−14} m^{2}/W, a saturable intensity of 10.12 GW/cm^{2} and a modulation depth of 61.2% at 800 nm under femto-second laser excitation. The measured value of the *effective nonlinear refractive index* decreases with increasing input power. Interestingly, it still remains large at the state of absorbance being saturated. This indicates that TI is highly favorable for the nonlinear optics application at high power regime (high nonlinear phase, low absorption).

More works are called for the investigation of the role of carrier dynamics in the slow time scale, and the value of *nonlinear refractive index* in the limit of defect-free topological insulators. It has been recently pointed out that the band-gap and Fermi energy level could be engineered in TI and the effect of band-gap tuning on the *nonlinear refractive index* remain unclear. More experiments with defect-free, thinner TI with controlled energy band structure will be required to further explore the third order nonlinear optics in the TI.

## Appendix

## A. Characterization of Bi_{2}Se_{3} sample

Topological insulator (TI) is a new class of quantum matter for which its band structure of surface state is similar to that of graphene, and it has attracted great attention in condensed-matter physics. Bi_{2}Se_{3} has been shown to be an ideal candidate for room-temperature topological insulating behavior as it has a band gap of 0.3 eV. We have synthesized the Bi_{2}Se_{3} NPs with the method introduced in Ref [23], and the Raman spectrum and AFM image have determined the identity and measured the thickness of the sample, respectively. High quality NPs exhibit hexagonal morphologies with planar dimensions. Figure 5
shows the SEM images of chemically grown nanonplatelets (NPs). It can be seen that NPs exhibit hexagonal morphologies with planar dimensions that could extend up to several micrometers.

## B. Laser beam characterization

A CCD was used to measure the beam waist from a Coherent femto-second laser. Firstly, we ensure that after the focusing objective, the laser beam axis is perpendicular to the pin hole of the CCD, which is mounted upon the translation stage. Then, by finely adjusting the position of the beam profiler through the Newport M-ILS250CC linear motorized stage, the laser beam intensity distribution before and after the focus point can be captured by the CCD. Figure 6
shows a typical laser spectrum, the beam intensity profile, beam waist near the focus point and the relation between laser beam intensity and the relative position of the focus point. The laser beam waist near the focus point is measured to be as broad as 40 *μ*m. The exact beam waist can be further inferred through fitting a typical *Z*-scan curve, which gives an estimated beam waist of 35 *μ*m.

## C.*Z*-scan measurement of CS_{2} sample

CS_{2} solution contained within a cuvette (1 mm in thickness) is deliberately used as a benchmark for calibration. Under the illumination of 76 GW/cm^{2} intensity at the focus point, the *Z*-scan measurements to CS_{2} sample are taken and the results are shown in the Fig. 7
. From the Fig. 7 (a), we can see that the open aperture measurement has a typical curve of optical limiting effect, which indicates a two-photon absorption (2PA) effect. Due to the existence of the nonlinear absorption, the closed aperture measurement shown in Fig. 7(b) is asymmetric. Dividing by the open aperture curve, the Fig. 7(c) trace exhibits a typical shape of *Z*-scan curve. The nonlinear on-axis phase shift ∆Φ is fitted to be 1.35 rad and the calculated *nonlinear refractive index* is about 2.66×10^{−19} m^{2}/W, which is in good agreement with (3±0.6)×10^{−19} m^{2}/W reported previously in Ref [25]. A plot of ∆Φ versus peak laser irradiance as measured from various *Z*-scan on the same CS_{2} cell is shown in Fig. 7(d). The linear behavior of this plot follows the theoretical model as derived for a cubic nonlinearity and the *nonlinear refractive index n*_{2} becomes saturable. By comparing the experimentally measured nonlinear optics value on the standard CS_{2} benchmark and the referenced value of CS_{2} in Ref [25], we are able to conclude that our current measurement system is reliable and pulse duration still remains at a reasonable value around 100 fs after passing through many optical elements.

## D. Fitting and calculation of the nonlinear refractive index

A closed aperture *Z*-scan trace shows the on-axis transmittance of the sample in the far field of a laser beam, when the sample is scanned through the focus of the beam. A sample with a *nonlinear refractive index n*_{2} acts as a thin lens in this setup. For instance, a sample with positive *n*_{2} will cause a divergence of the beam at pre-focal positions and a convergence at post-focal positions. This result is a typical valley-peak shape of the closed aperture trace (Fig. 8(b)
). To calculate the on-axis phase shift ∆Φ, we used the Gaussian decomposition method, and expanded the calculations up to the third order in ∆Φ, as shown in Eq. (1). While the first order approximation is only allowed to determine small phase shifts, the third order approximation is valid up to ∆Φ ≈1.75 [26].

To examine the effect of the quartz substrate, a pure quartz plate (1 mm in thickness) was taken as a reference for the open aperture and closed aperture measurement, at the intensity of 10.4 GW/cm^{2}. The results are shown in the Fig. 8, and we can see that the quartz sample has little change in normalized transmittance. In contrast with the curve of the Bi_{2}Se_{3} sample, the substrate effect can be safely ignored. Using the third order approximation to fit the experimental data, the ∆Φ ≈ 1.1 rad and the *nonlinear refractive index* is calculated to be *n*_{2} = 2.26×10^{−14} m^{2}/W. Therefore, we could confirm that the peak-valley variation is only contributed by the third order nonlinear optics effect from the TI sample, and safely exclude the effect of the quartz substrate.

## Acknowledgment

This work is partially supported by the National 973 Program of China (Grant No. 2012CB315701), the National Natural Science Foundation of China (Grant Nos. 61025024 and 61205125), the National 863 Program of China (Grant No. 2011AA010203). Han ZHANG acknowledges the support from Program for New Century Excellent Talents in University of China (Grant No. NCET 11-0135), “Thousand Talents Program” for Distinguished Young Scholars and National Natural Science Fund Foundation of China for Excellent Young Scholars (Grant No. 61222505).

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