1d) shows a typical metallic behavior and saturates below ~30 K 27, 28. Figure 1d, e show the basic electrical characterization. Multiple Hall bar devices with current channels along different crystalline directions (Fig. Thus, the induction of BCD via breaking the threefold rotational symmetry 26 does not occur in Bi 2Se 3 films, as confirmed by our angle dependent transport measurements below. In addition, the lattice constant of Bi 2Se 3 film relaxes to its bulk value, implying the absence of strain from the substrate 25. The first quintuple layer (QL) of Bi 2Se 3 is completely relaxed by van der Waals bonds 25. High-quality Bi 2Se 3 films were grown on Al 2O 3 (0001) substrates in a molecular beam epitaxy system. 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. 3D TIs are ideal platforms in searching for extrinsic electrical SHG in the absence of a BCD. 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. 1b) 3 thus, the BCD-induced nonlinear Hall effect is not allowed. Furthermore, threefold rotational symmetry of the TI surface in Fig. With an inversion-symmetric bulk, 3D TIs such as Bi 2Se 3, Bi 2Te 3, and Sb 2Te 3 host electrical SHG only on the surfaces. 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. However, the electrical SHG has not explored in surface/interface systems with time-reversal symmetry. Inversion symmetry is absent in low-symmetry crystals (such as WTe 2 5, 6, 10), and on a surface or an interface.
![fourier transform of thomas fermi screening fourier transform of thomas fermi screening](https://i.ytimg.com/vi/BOyQENLLeZM/maxresdefault.jpg)
Therefore, a search for electrical SHG independent of the BCD is desirable. 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. Electrical second-harmonic generation (SHG), including the nonlinear Hall effect, can exist only when a system lacks inversion symmetry 7, 8, 9. The BCD generates an effective magnetic field in a stationary state, thus leading to the nonlinear Hall effect 3. The nonlinear Hall effect due to the BCD was observed recently in bilayer and few-layer WTe 2 5, 6. A quantum origin of the nonlinear Hall effect in time-reversal-invariant materials is the Berry curvature dipole (BCD) 3.
![fourier transform of thomas fermi screening fourier transform of thomas fermi screening](https://royalsocietypublishing.org/cms/asset/17b63523-3bc6-459d-8eaf-bb732a5cb133/rspa20160258f07.jpg)
![fourier transform of thomas fermi screening fourier transform of thomas fermi screening](http://media.cheggcdn.com/media/f9b/f9b792d9-c6de-4bbb-884e-4e23ac767b69/php9VpSZp.png)
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. 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 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 time-reversal symmetry breaking.