Abstract
The nonlinear Hall effect due to Berry curvature dipole (BCD) induces frequency doubling, which was recently observed in timereversalinvariant materials. Here we report novel electric frequency doubling in the absence of BCD on a surface of the topological insulator Bi_{2}Se_{3} under zero magnetic field. We observe that the frequencydoubling voltage transverse to the applied ac current shows a threefold rotational symmetry, whereas it forbids BCD. One of the mechanisms compatible with the symmetry is skew scattering, arising from the inherent chirality of the topological surface state. We introduce the Berry curvature triple, a highorder moment of the Berry curvature, to explain skew scattering under the threefold rotational symmetry. Our work paves the way to obtain a giant secondorder nonlinear electric effect in high mobility quantum materials, as the skew scattering surpasses other mechanisms in the clean limit.
Introduction
The Hall effect, the generation of voltage transverse to an electric current and a magnetic field, and the anomalous Hall effect (AHE) in magnetic materials^{1} require timereversal symmetry breaking. These effects refer to a transverse electric response in the linear region, where the Hall voltage V_{y} scales linearly with the longitudinal current I_{x}. The secondorder (nonlinear) Hall effect, in which V_{y} depends quadratically on I_{x}, has attracted attention in condensed matter physics^{2,3,4}. A quantum origin of the nonlinear Hall effect in timereversalinvariant materials is the Berry curvature dipole (BCD)^{3}. The nonlinear Hall effect due to the BCD was observed recently in bilayer and fewlayer WTe_{2}^{5,6}. The BCD generates an effective magnetic field in a stationary state, thus leading to the nonlinear Hall effect^{3}. Electrical secondharmonic generation (SHG), including the nonlinear Hall effect, can exist only when a system lacks inversion symmetry^{7,8,9}. Despite growing interest of BCD^{10,11,12,13,14}, it is subject to strict crystal symmetry restrictions and vanishes in certain crystals even without inversion symmetry^{3}, while secondorder response is still allowed. Therefore, a search for electrical SHG independent of the BCD is desirable.
Inversion symmetry is absent in lowsymmetry crystals (such as WTe_{2}^{5,6,10}), and on a surface or an interface. However, the electrical SHG has not explored in surface/interface systems with timereversal symmetry. Threedimensional (3D) topological insulators (TIs) have attracted great interest due to the topological surface state (TSS) with spinmomentum locking^{15,16,17} for applications in spintronics and quantum computing^{18,19,20}. With an inversionsymmetric bulk, 3D TIs such as Bi_{2}Se_{3}, Bi_{2}Te_{3}, and Sb_{2}Te_{3} host electrical SHG only on the surfaces. Furthermore, threefold rotational symmetry of the TI surface in Fig. 1a forces a BCD to vanish (Fig. 1b)^{3}; thus, the BCDinduced nonlinear Hall effect is not allowed. In addition to the intrinsic contribution by a BCD, extrinsic effects arising from impurity or phonon scatterings, as intensively studied in AHE^{1}, are yet to be well sorted out for nonlinear effects. 3D TIs are ideal platforms in searching for extrinsic electrical SHG in the absence of a BCD. While recent theoretical studies addressed extrinsic mechanisms^{21,22,23,24}, an experimental observation of extrinsic contributions to the electrical SHG has not been reported.
In this work, we show the observation of electrical SHG in the 3D TI Bi_{2}Se_{3} with timereversal symmetry. The transverse voltage response depends quadratically on the applied current in the nonmagnetic Bi_{2}Se_{3} films under zero magnetic field. The observed secondorder response follows a threefold rotational symmetry on the surface of Bi_{2}Se_{3}. Notably, the symmetry excludes a BCD, which distinguishes the mechanism for electrical SHG from the previous studies^{5,6}. We consider our observation arising dominantly from skew scattering in the TSS with its inherently chiral wave function. Instead of a BCD, we introduce the Berry curvature triple, which quantifies the moment of the Berry curvature under the threefold rotational symmetry. The skew scattering mechanism applies to a much wider class of noncentrosymmetric materials as broken inversion is the only symmetry constraint unlike the BCD.
Results
Observation of electric SHG
Highquality Bi_{2}Se_{3} films were grown on Al_{2}O_{3} (0001) substrates in a molecular beam epitaxy system. The first quintuple layer (QL) of Bi_{2}Se_{3} is completely relaxed by van der Waals bonds^{25}. In addition, the lattice constant of Bi_{2}Se_{3} film relaxes to its bulk value, implying the absence of strain from the substrate^{25}. Thus, the induction of BCD via breaking the threefold rotational symmetry^{26} does not occur in Bi_{2}Se_{3} films, as confirmed by our angle dependent transport measurements below. Multiple Hall bar devices with current channels along different crystalline directions (Fig. 1c) were fabricated. Figure 1d, e show the basic electrical characterization. The longitudinal resistivity ρ (Fig. 1d) shows a typical metallic behavior and saturates below ~30 K^{27,28}. Figure 1e displays the longitudinal R_{xx} and Hall R_{yx} resistances as a function of an outofplane magnetic field at 2 K. R_{xx} at the low field region exhibits the effect of weak antilocalization, indicative of 2D surface transports^{29}. R_{yx} depends linearly on the magnetic field, from which the ntype carrier density n_{2D} is extracted to be ~6.26 × 10^{13} cm^{−2}. n_{2D} changes < 2.3% for temperature (T) of 2 < T < 300 K.
To explore the nonlinear electric transport, we perform harmonic measurements using lowfrequency lockin techniques schematically shown in Fig. 2a. We apply the ac current I_{x}(t) = Isinωt along the x direction and measure the voltage V_{y} perpendicular to the current. Under timereversal and threefold rotational symmetries, the transverse voltage response does not contain the linear contribution, leading to the expression
which contains the SHG signal \({{V}}_{{y}}^{2\omega } = \frac{1}{2}{{R}}_{{{yxx}}}^{\left( 2 \right)}{{I}}^2\sin \left( {2\omega {{t}}  {\pi}/2} \right)\). Note that the coefficient \({{R}}_{{{yxx}}}^{\left( 2 \right)}\) is proportional to the secondorder conductivity \({\sigma}_{{{yxx}}}^{\left( 2 \right)}\) (see Supplementary Note 1), which can be finite in noncentrosymmetric materials^{3}.
Figure 2b shows the second harmonic transverse voltage under zero magnetic field in 20 QL Bi_{2}Se_{3}. Its quadratic dependence on the ac current (\({{V}}_{{y}}^{2{\omega}} \propto {{I}}^2\)) reveals the electrical SHG from a timereversalinvariant 3D TI. Equivalently, the second harmonic transverse resistance defined as \({{R}}_{{{yx}}}^{2{\omega}} \equiv {{V}}_{{y}}^{2{\omega}}/{{I}}\) scales linearly with I (Fig. 2c). Moreover, it changes the sign when we invert the current direction and the corresponding Hall probes (schematic in the inset of Fig. 2c). This is consistent with the secondorder nature of nonlinear transport in Eq. (1). The electric SHG has little dependence on the input frequencies ranging from 9 to 263 Hz (see Supplementary Fig. 1).
Figure 2d displays the \({{R}}_{{{yx}}}^{2{\omega}}\left( {{I}} \right)\) data at different temperatures. The slope of \({{R}}_{{{yx}}}^{2{\omega}}\left( {{I}} \right)\) (i.e. \({{R}}_{{{yxx}}}^{\left( 2 \right)}\)) quantifies the magnitude of the electrical SHG. \({{R}}_{{{yxx}}}^{\left( 2 \right)}\) decreases gradually as temperature increases in Fig. 2e. In general, finite temperature affects the nonlinear electric transport through thermal smearing of the electron distribution function f and the change of the electron scattering time τ. Thermal smearing has little effect on the result as the Fermi energy is much higher than thermal energy k_{B}T in our Bi_{2}Se_{3} (k_{B}: the Boltzmann constant). To reveal the effect of τ, we depict the measured carrier mobility µ in Fig. 2f. Both the SHG signal and mobility tend to decrease as temperature rises.
Angular dependence and scaling of nonlinear transport
To characterize the angular dependence of nonlinear electric transport, we measure various devices with the current applied along different crystal directions on 20 QL Bi_{2}Se_{3} (Fig. 1c). The current direction is denoted by angle Θ with respect to the \(\overline \Gamma \overline K\) direction (i.e., [−1, 1, 0] direction on Bi_{2}Se_{3} (111) surface of the primitive lattice in real space) in Fig. 3. \({{R}}_{{{yx}}}^{2{\omega}}\) shows the maximum value when the current direction is along \(\overline \Gamma \overline K\) (Fig. 2b, c), and decreases when the current is rotated 15° away from \(\overline \Gamma \overline K\) in Fig. 3a. For Θ = 30, i.e., with the ac current along the \(\overline \Gamma \overline M\) direction, \({{R}}_{{{yx}}}^{2{\omega}}\) becomes vanishingly small (Fig. 3b). \({{R}}_{{{yx}}}^{2{\omega}}\) switches sign with a similar magnitude when the current direction is rotated by 60° from the \(\overline \Gamma \overline K\) to \(\overline \Gamma \overline K ^\prime\) direction in Fig. 3c. The small nonsymmetry of \({{R}}_{{{yx}}}^{2{\omega}}\left( {{I}} \right)\) at the positive and negative current in Fig. 3a–c can be due to misalignments of Hall bar. The electric SHG measured at 24 different directions is summarized in Fig. 3d, which shows the threefold angular dependence of \({{R}}_{{{yxx}}}^{\left( 2 \right)}\). The similar angular dependence is also observed in 10 QL Bi_{2}Se_{3} (Supplementary Fig. 2). We emphasize that threefold rotational symmetric signal with sign change excludes the Joule heating effect as an origin, which is isotropic and generally leads to the third harmonic generation. The threefold symmetry also excludes a BCD, while the helical spin texture^{30} and the Berry curvature^{31} (Fig. 1b) on the hexagonally warped Fermi surface (FS) of the TSS^{32,33} share the same angular dependence. We note that the Berry curvature has the opposite sign along \(\overline \Gamma \overline K\) and \(\overline \Gamma \overline K ^\prime\) due to timereversal symmetry.
The nontrivial wavefunction on the TSS with scattering by impurities or phonons can give rise to finite electrical SHG^{24}. To investigate the microscopic mechanism, we examine the scaling properties of the secondorder transport with respect to the linear conductivity σ of the film using the data in Figs. 1d and 2e. Figure 4a shows that the experimental data fit well with \(\frac{{E_y^{\left( 2 \right)}}}{{E_x^{\left( 2 \right)}}} = a\sigma ^2 + b\), where \(E_y^{\left( 2 \right)} = \frac{{V_y^{2\omega }}}{W}\) and \(E_x = \frac{{V_x^\omega }}{L}\) (W and L are the width and length of the sample, respectively). The linear and secondorder conductivities σ and \(\sigma _{yxx}^{\left( 2 \right)}\) are related by \(J_y^{\left( 2 \right)} = \sigma _{yxx}^{\left( 2 \right)}E_x^2 = \sigma E_y^{\left( 2 \right)}\), so the coefficients a and b represent contributions in \(\sigma _{yxx}^{\left( 2 \right)}\) that scale as σ^{3} and σ, respectively. Furthermore, σ is proportional to τ for low frequencies compared to τ^{−1}. Therefore, the intercept b amounts to the τ linear contributions of the secondorder conductivity, which are generally attributed to BCD^{3} and/or side jump^{6}. Note that the former is absent in our case for the symmetry reason, so we attribute the τlinear contribution to side jump. On the other hand, the slope a quantifies the contribution \(\sigma _{yxx}^{\left( 2 \right)} \propto \tau ^3\), which originates from skew scattering as we discuss below. We obtain similar fitting results for \(\Theta\) = 15° in Fig. 4b and also in 10 QL Bi_{2}Se_{3} (see Supplementary Fig. 2). Notably, the cubic contribution plays a dominant role over the linear one as σ increases in Bi_{2}Se_{3}, and these two contributions are of opposite signs as shown in Fig. 4a, b and are separated in Supplementary Fig. 3. The scaling of electrical SHG with respect to the surface linear conductivity σ_{s} is also analyzed in Supplementary Fig. 4.
Physical origin of nonlinear transport
The TI Bi_{2}Se_{3} possesses timereversal and inversion symmetries in the bulk. However, inversion is broken on the surface and hence the metallic TSS with C_{3v} crystalline symmetry can host electrical SHG. It takes the form^{24,34}
where Θ is the angle of the applied electric field E with respect to the \(\overline \Gamma \overline K\) direction and the current density J is measured perpendicular to E. There is only one independent element σ^{(2)} in the secondorder conductivity tensor \({\sigma}_{{\mathrm{abc}}}^{\left( 2 \right)}\) for a twodimensional system with C_{3v} symmetry (see Methods).
Skew scattering is one of the microscopic mechanisms that contributes to σ^{(2)}. It arises even classically when there are nontrivial impurity potentials lacking inversion on the atomic scale^{8,34,35} or by local correlation of spins^{36}. Alternatively, without relying details of impurities, quantum Bloch functions can imprint inversion breaking and trigger skew scattering, which is the case for the TSS^{24,34}. There is a semiclassical picture for skew scattering in a secondorder process, schematically depicted in Fig. 4c. The hexagonally warped Fermi surface consists of the positive and negative Berry curvature segments. Since both segments are anisotropic, they acquire finite but opposite velocities in the secondorder response. When we construct a wave packet from states on the Fermi surface, it selfrotates due to finite Berry curvature and the rotation direction depends on the sign of Berry curvature. Like the Magnus effect, even an isotropic scatterer deflects the motion of wave packets in a preferred direction due to the selfrotation, thus leading to finite SHG.
The semiclassical Boltzmann transport calculation^{24} based on the model Hamiltonian of TSS^{32,37} leads to the linear conductivity from the TSS \(\sigma _{{\mathrm{TSS}}} = \frac{{e^2\tau \it{\epsilon} _F}}{{4\pi {\hbar}^{2}}}\) and the secondorder conductivity from skew scattering is given by \(\sigma ^{\left( 2 \right)} = \frac{{e^{3} v\tau ^{3}}}{{{\hbar}^2 \widetilde{\tau} }}\), where τ is the transport scattering time, \(\widetilde \tau\) is the skew scattering time, e is the electric charge, ∈_{F} is the Fermi energy, and v is the Dirac velocity. Importantly, skew scattering yields σ^{(2)} ∝ τ^{3} (assuming that \(\widetilde \tau\) is constant) while other contributions including side jump have weaker powers in τ, which distinguishes the skew scattering contribution. The experimentally observed \(\sigma _{yxx}^{\left( 2 \right)} \propto \sigma ^3\) behavior is supported by the skew scattering mechanism, whose contribution is the largest in our observations.
The secondorder conductivity obeys the surface crystalline symmetry to have the form \(\sigma _{yxx}^{\left( 2 \right)} = \sigma ^{\left( 2 \right)}\cos 3{{\Theta }},\) according to Eq. (2) (Fig. 4d), which is in agreement with our experiment. Instead of a BCD, the threefold rotational symmetry inspires us to define the Berry curvature triple T, a higherorder moment of the Berry curvature distribution in the momentum space. It quantifies the strength of the Berry curvature on the Fermi surface, respecting threefold rotation: \(T\left( {{\it{\epsilon }}_F} \right) = 2\pi {\hbar} {\int} {\frac{{d^2k}}{{\left( {2\pi } \right)^2}}} \delta \left( {{\it{\epsilon }}_F  {\it{\epsilon }}_{\mathbf{k}}} \right){{\Omega }}_z\left( {\mathbf{k}} \right)\cos 3\theta _{\mathbf{k}}\) (θ_{k}: the angle measured from the \(\overline \Gamma \overline K\) line). For the TSS, we obtain \(T\left( {{\it{\epsilon }}_F} \right) = \frac{{\lambda {\it{\epsilon }}_F}}{{2{\hbar}^{2} v^{3}}}\). The Berry curvature triple is related to the skew scattering time \(\widetilde \tau\). When we consider unscreened Coulomb impurities with the strength characterized by the dimensionless parameter \(\alpha = \frac{{e^2Q}}{{4\pi \varepsilon _0\varepsilon {\hbar} v}}\), where Q is the impurity charge, ε_{0} is the vacuum permittivity, and ε is the dielectric constant, we find \(\widetilde \tau \approx 4\pi ^2n_i\alpha ^3v^2T\left( {{\it{\epsilon }}_F} \right)\) (see Supplementary Note 2).
We now provide the theoretical estimate of the secondorder response from skew scattering. Though the secondorder response arises only on the surface, both 2D surface and bulk states contribute to σ. As the contribution from the TSS is ~40% from the top and bottom surfaces^{38}, we estimate τ ≈ 0.1 ps and \(\widetilde \tau \approx 10\,{\mathrm{ps}}\) (see Methods section). The ratio \(\tau /\widetilde \tau\) of ~1% quantifies the relative strength of skew scattering. The estimated τ and \(\widetilde \tau\) result in the theoretical value σ^{(2)} ≈ 1.0 × 10^{−11} A·V^{−2}·m. This is about three times larger than the experimentally observed value σ^{(2)} = 2.9 × 10^{−12} A·V^{−2}·m. We can attribute this difference to the partial cancellation of the secondorder response; the contribution of the top surface is dominant over that of the bottom surface. In addition, screening of the Coulomb interaction reduces the response (see Supplementary Fig. 5).
Discussion
We have demonstrated the electric SHG in a nonmagnetic 3D TI under zero magnetic field. It provides an example of BCDindependent nonlinear transverse transport, which is further revealed to arise from skew scattering. This skew scattering mechanism can be applicable to a broader class of noncentrosymmetric quantum materials, utilizing the chirality of electron wavefunction in Weyl and Dirac fermions^{39}. Though our work reveals the nonlinear transport under low frequencies, it can be extended to higher frequency regimes such as GHz and THz. Thus, the electric SHG is complementary to previous optoelectronic approaches^{34,40} to reveal the underlying physics of nonlinear effects.
Berry curvature is allowed to exist in the TSS^{31,41}, and concentrates in regions around \(\overline K\) \(\left( {\overline K ^\prime } \right)\) points in Fig. 1b, leading to finite Berry curvature triple. Finite Berry curvature also affects the electron distribution function through the collision integral and the anomalous and side jump velocities^{24}. The intrinsic contribution due to the anomalous velocity and hence BCD is absent in Bi_{2}Se_{3} due to the symmetry reason^{3}; however, the extrinsic contributions such as skew scattering and side jump persist^{21}. The skew scattering contribution dominates in the weak impurity limit (τ → ∞)^{23,24} because of its highorder τ dependence. Though a full quantitative understanding of various contributions to nonlinear electric transports remains elusive^{21} which may include phonons, domain boundaries, impurities, and Berry curvature^{42}, identifying major mechanisms is an important step not only for the fundamental understanding of underlying principle, but for the development of rectification or secondharmonic devices for energy harvesting and highfrequency communication. The extrinsic nonlinear effect observed in Bi_{2}Se_{3} is comparable in magnitude to the intrinsic one in fewlayer WTe_{2}^{6}, which has a 2D nonlinear conductivity of ~10^{−12} A·V^{−2}·m. Moreover, the extrinsic mechanism exemplified here applies to a wider class of materials with inversionsymmetry breaking, such as graphene/hexagonalboronnitride heterostructures^{43}, Dirac semimetal ZrTe_{5}^{44,45} and the twodimensional electron gas at the LaAlO_{3}/SrTiO_{3} interface^{46}. Engineering scattering processes in above materials is a promising way to achieve a prominent SHG by utilizing their much higher carrier mobilities. A higher mobility and long scattering time improve the efficiency in device applications since skew scattering has a higher order dependence on τ^{1,24,47}.
Methods
Sample preparation and electric measurements
Bi_{2}Se_{3} films were grown on Al_{2}O_{3} (0001) substrates in a molecular beam epitaxy system with a base pressure < 2 × 10^{−9} mbar, as detailed in Tian et al.^{47}. Van der Waals epitaxy of Bi_{2}Se_{3} film was achieved by adopting the twostep growth method^{25,27,48,49}. For transport measurements, a capping layer of MgO (2 nm)/Al_{2}O_{3} (3 nm) was deposited on top of the films prior to device fabrication. Hall bar devices were fabricated using the standard photolithography and Argon plasma etching. They were wirebonded to the sample holder and installed in a physical property measurement system (PPMS, Quantum Design) for transport measurements. We performed lowfrequency ac harmonic electric measurements, using Keithley 6221 current sources and Stanford Research SR830 lockin amplifiers. During the measurements, a sinusoidal current with a constant amplitude and certain frequency is applied to the devices, and the inphase first harmonic V_{ω} and outofphase second harmonic V_{2ω} longitudinal and transverse voltages were measured simultaneously by four lockin amplifiers.
Theoretical modeling and estimate
The Hamiltonian for the TSS is^{32,37}
where k_{±} = k_{x} ± ik_{y}, σ_{a} denotes the Pauli matrix (a = x,y,z), and λ quantifies the hexagonal warping^{32}. In this section, the x axis is set perpendicular to the reflection plane, i.e., along the \(\overline {{\Gamma }} \overline {\mathrm{K}}\) line. For the surface state of Bi_{2}Se_{3}, we find v = 5 × 10^{5} m/s and λ = 80 eV·Å^{3}, and the FS is located above the Dirac point, where a hexagonally warped FS was found^{30,33}.
In general, the current response quadratic to the electric field E takes the form \(J_a^{\left( 2 \right)} = \sigma _{abc}^{\left( 2 \right)}E_bE_c\), where \(\sigma _{abc}^{\left( 2 \right)}\) is the secondorder conductivity. For a twodimensional system with C_{3V} symmetry like the TSS, it has only one independent element \(\sigma ^{\left( 2 \right)} \equiv \sigma _{xxy}^{\left( 2 \right)} = \sigma _{xyx}^{\left( 2 \right)} = \sigma _{yxx}^{\left( 2 \right)} =  \sigma _{yyy}^{\left( 2 \right)}\). To estimate the transport properties, we assume Coulomb impurities, randomly distributed in a sample. Taking account of the ThomasFermi screening, we write the Fourier transform of the Coulomb interaction as \(V\left( q \right) = \frac{{2\pi \alpha {\hbar} v}}{{q + q_{{\mathrm{TF}}}}}\), where q_{TF} is the ThomasFermi wavevector. Here, we consider unscreened Coulomb impurities (q_{TF} = 0), which we discuss below.
In estimating τ and \(\widetilde \tau\), we use the dielectric constant^{50} ∈ ≈ 100, leading to \(\alpha \approx \frac{1}{{23}}\). We use the previous observation that the contribution of the TSS from the top and bottom surfaces to the total conduction is ~40%^{38} and assume that the impurity density n_{i} is approximately the same as the carrier density n_{2D}. Thus, the observed linear conductivity σ = 2.5 × 10^{−3}Ω^{−1} at 10 K leads to the carrier density of the TSS n_{TSS} = 2.43 × 10^{12} cm^{−2}, the corresponding Fermi wavelength \(\lambda _F = \sqrt {\frac{\pi }{{n_{{\mathrm{TSS}}}}}} = 11.4\,{\mathrm{nm}}\), the scattering time τ ≈ 0.1 ps, and the skew scattering time \(\widetilde \tau \approx 10\,{\mathrm{ps}}\), where we use the expressions^{24} \(\sigma _{{\mathrm{TSS}}} = \frac{{e^2\tau {\it{\epsilon }}_F}}{{4\pi {\hbar}^{2}}}\), \(\tau ^{  1} = \frac{\pi }{2} n_{i} \alpha ^{2} v {\lambda}_{F}\), and \({\widetilde{\tau}}^{1} = \frac{{4\pi ^3}}{{\hbar}}\frac{{n_{i}\alpha^{3}\lambda }}{{\lambda _{F}}}\). The small ratio of \(\frac{{\uptau }}{{\widetilde \tau }} \ll 1\) satisfies the condition of the perturbative treatment of impurities in the semiclassical Boltzmann theory.
The ThomasFermi wavelength \(\lambda _{{\mathrm{TF}}} = \frac{{2\pi }}{{q_{{\mathrm{TF}}}}}\) is typically ranging from 26 to 90 nm^{51,52}, resulting in the ratio \(\lambda _F/\lambda _{{\mathrm{TF}}} \ \lesssim\ 0.4\). We describe the detailed calculations and discussion about the effect of screening in Supplementary Note 2 and Supplementary Fig. 5. We note that for shortrange impurities or in the strong screening limit, i.e., λ_{TF} → 0, skew scattering vanishes in a gapless Dirac system^{24,34}.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank Steven S.L. Zhang and G. Vignale for discussions. The work was partially supported by SpOTLITE program (A*STAR grant, A18A6b0057) through RIE2020 funds, and Singapore Ministry of Education (MOE) Tier 1 (R263000D61114). P.H. acknowledges the startup funding from Fudan University.
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P.H. fabricated the devices, performed transport measurements, and analyzed the data. D.Z. grew the films. H.I. and L.F. performed theoretical studies. C.H.H. contributed to Berry curvature calculation. All authors discussed the results. P.H., H.I., L.F., and H.Y. wrote the manuscript.
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He, P., Isobe, H., Zhu, D. et al. Quantum frequency doubling in the topological insulator Bi_{2}Se_{3}. Nat Commun 12, 698 (2021). https://doi.org/10.1038/s41467021209831
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DOI: https://doi.org/10.1038/s41467021209831
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