Quantum frequency doubling in the topological insulator Bi2Se3

The nonlinear Hall effect due to Berry curvature dipole (BCD) induces frequency doubling, which was recently observed in time-reversal-invariant materials. Here we report novel electric frequency doubling in the absence of BCD on a surface of the topological insulator Bi2Se3 under zero magnetic field. We observe that the frequency-doubling 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 high-order moment of the Berry curvature, to explain skew scattering under the threefold rotational symmetry. Our work paves the way to obtain a giant second-order nonlinear electric effect in high mobility quantum materials, as the skew scattering surpasses other mechanisms in the clean limit. Berry curvature dipole (BCD) leads to the nonlinear Hall effect manifested as a frequency doubling in topological materials. Here, the authors report electric frequency doubling in the absence of BCD and magnetic field on a surface of Bi2Se3 due to skew scattering arising from inherent chirality of the topological surface states.

T he 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 second-order (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 time-reversal-invariant materials is the Berry curvature dipole (BCD) 3 . The nonlinear Hall effect due to the BCD was observed recently in bilayer and few-layer WTe 2 5,6 . The BCD generates an effective magnetic field in a stationary state, thus leading to the nonlinear Hall effect 3 . Electrical second-harmonic 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 second-order response is still allowed. Therefore, a search for electrical SHG independent of the BCD is desirable.
Inversion symmetry is absent in low-symmetry 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 time-reversal symmetry. Three-dimensional (3D) topological insulators (TIs) have attracted great interest due to the topological surface state (TSS) with spin-momentum locking [15][16][17] for applications in spintronics and quantum computing [18][19][20] . With an inversion-symmetric 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 BCD-induced 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 time-reversal 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 second-order 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. High-quality 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 outof-plane magnetic field at 2 K. R xx at the low field region exhibits the effect of weak anti-localization, indicative of 2D surface transports 29 . R yx depends linearly on the magnetic field, from which the n-type 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 low-frequency lock-in 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 time-reversal and threefold rotational symmetries, the transverse voltage response does not contain the linear contribution, leading to the expression which contains the SHG signal V 2ω Note that the coefficient R 2 ð Þ yxx is proportional to the second-order conductivity σ 2 ð Þ yxx (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 2ω y / I 2 ) reveals the electrical SHG from a timereversal-invariant 3D TI. Equivalently, the second harmonic transverse resistance defined as R 2ω yx V 2ω y =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 second-order 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 2ω yx I ð Þ data at different temperatures. The slope of R 2ω yx I ð Þ (i.e. R 2 ð Þ yxx ) quantifies the magnitude of the electrical SHG. R 2 ð Þ yxx 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 ΓK 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 2ω yx shows the maximum value when the current direction is along ΓK (Fig. 2b, c), and decreases when the current is rotated 15°away from ΓK in Fig. 3a. For Θ = 30, i.e., with the ac current along the ΓM direction, R 2ω yx becomes vanishingly small (Fig. 3b). R 2ω yx switches sign with a similar  Fig. 3d, which shows the threefold angular dependence of R 2 ð Þ yxx . 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 ΓK and ΓK 0 due to time-reversal 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 second-order 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 y , so the coefficients a and b represent contributions in σ 2 ð Þ yxx 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 second-order 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 τ-linear contribution to side jump. On the other hand, the slope a quantifies the contribution σ 2 ð Þ yxx / τ 3 , which originates from skew scattering as we discuss below. We obtain similar fitting results for Θ = 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 time-reversal 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 ΓK direction and the current density J is measured perpendicular to E. There is only one independent element σ (2) in the second-order conductivity tensor σ 2 ð Þ abc for a two-dimensional 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 second-order 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 second-order response. When we construct a wave packet from states on the Fermi surface, it self-rotates 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 self-rotation, 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 σ TSS ¼ e 2 τϵ F 4π h 2 and the second-order conductivity from skew scattering is given by σ 2 ð Þ ¼ e 3 vτ 3 h 2 e τ , where τ is the transport scattering time, e τ 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 e τ is constant) while other contributions including side jump have weaker powers in τ, which distinguishes the skew scattering contribution. The experimentally observed σ 2 ð Þ yxx / σ 3 behavior is supported by the skew scattering mechanism, whose contribution is the largest in our observations.
The second-order conductivity obeys the surface crystalline symmetry to have the form σ 2 ð Þ yxx ¼ σ 2 ð Þ cos 3Θ; 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 higher-order 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: (θ k : the angle measured from the ΓK line). For the TSS, we The Berry curvature triple is related to the skew scattering time e τ. When we consider unscreened Coulomb impurities with the strength characterized by the dimensionless parameter α ¼ e 2 Q 4πε 0 ε hv , where Q is the impurity charge, ε 0 is the vacuum permittivity, and ε is the dielectric constant, we find e τ % 4π 2 n i α 3 v 2 T ϵ F ð Þ (see Supplementary Note 2). We now provide the theoretical estimate of the second-order response from skew scattering. Though the second-order 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 e τ % 10 ps (see Methods section). The ratio τ=e τ of~1% quantifies the relative strength of skew scattering. The estimated τ and e τ 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 second-order 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 K K 0 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 high-order 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 second-harmonic devices for energy harvesting and high-frequency communication. The extrinsic nonlinear effect observed in Bi 2 Se 3 is comparable in magnitude to the intrinsic one in few-layer 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 inversion-symmetry breaking, such as graphene/hexagonal-boron-nitride heterostructures 43 , Dirac semimetal ZrTe 5 44,45 and the two-dimensional 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 two-step 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 wire-bonded to the sample holder and installed in a physical property measurement system (PPMS, Quantum Design) for transport measurements. We performed low-frequency ac harmonic electric measurements, using Keithley 6221 current sources and Stanford Research SR830 lock-in amplifiers. During the measurements, a sinusoidal current with a constant amplitude and certain frequency is applied to the devices, and the in-phase first harmonic V ω and out-of-phase second harmonic V 2ω longitudinal and transverse voltages were measured simultaneously by four lock-in 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 ΓK 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 abc is the second-order conductivity. For a twodimensional system with C 3V symmetry like the TSS, it has only one independent element σ 2 ð Þ σ 2 ð Þ yyy . To estimate the transport properties, we assume Coulomb impurities, randomly distributed in a sample. Taking account of the Thomas-Fermi screening, we write the Fourier transform of the Coulomb interaction as V q ð Þ ¼ 2πα hv qþq TF , where q TF is the Thomas-Fermi wavevector. Here, we consider unscreened Coulomb impurities (q TF = 0), which we discuss below.
In estimating τ and e τ, we use the dielectric constant 50 ∈ ≈ 100, leading to α % 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 λ F ¼ ffiffiffiffiffiffi π n TSS q ¼ 11:4 nm, the scattering time τ ≈ 0.1 ps, and the skew scattering time e τ % 10 ps, where we use the expressions 24 σ TSS ¼ e 2 τϵ F 4π h 2 , τ À1 ¼ π 2 n i α 2 vλ F , and e τ À1 ¼ 4π 3 h n i α 3 λ λ F . The small ratio of τ e τ ( 1 satisfies the condition of the perturbative treatment of impurities in the semiclassical Boltzmann theory. The Thomas-Fermi wavelength λ TF ¼ 2π q TF is typically ranging from 26 to 90 nm 51,52 , resulting in the ratio λ F =λ TF ≲ 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 short-range 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.