Nonlinear optical Hall effect in Weyl semimetal WTe2

The ordinary Hall effect refers to generation of a transverse voltage upon exertion of an electric field in the presence of an out-of-plane magnetic field. While a linear Hall effect is commonly observed in systems with breaking time-reversal symmetry via an applied external magnetic field or their intrinsic magnetization1, 2, a nonlinear Hall effect can generically occur in non-magnetic systems associated with a nonvanishing Berry curvature dipole3. Here we report, observations of a nonlinear optical Hall effect in a Weyl semimetal WTe2 without an applied magnetic field at room temperature. We observe an optical Hall effect resulting in a polarization rotation of the reflected light, referred to as the nonlinear Kerr rotation. The nonlinear Kerr rotation linearly depends on the charge current and optical power, which manifests the fourth-order nonlinearity. We quantitatively determine the fourth-order susceptibility, which exhibits strong anisotropy depending on the directions of the charge current and the light polarization. Employing symmetry analysis of Berry curvature multipoles, we demonstrate that the nonlinear Kerr rotations can arise from the Berry curvature hexapole allowed by the crystalline symmetries of WTe2. There also exist marginal signals that are incompatible with the symmetries, which suggest a hidden phase associated with the nonlinear process.

Whereas the ordinary Hall effect of electrons discovered in 1879 by Edwin H. Hall commonly arises in a two-dimensional conductor under a perpendicular magnetic field, the Hall effect has become a ubiquitous phenomenon, occurring in various systems ranging from magnetic insulators to non-magnetic semiconductors 1,2 . Notably, the nonlinear Hall effect has been theoretically predicted and experimentally observed in a non-magnetic system under a timereversal symmetric condition [3][4][5][6] . This finding stands a contrast to the a priori sense that the Hall effect should occur without time-reversal symmetry, since time-reversal symmetry zeroes the Chern number, which dictates the quantized Hall conductance of the two-dimensional electrons 7,8 . The key innovation was to reveal that it is only in the first-order response regime that time-reversal symmetry suppresses the Hall voltage, in which the Hall voltage is linearly proportional to a driving electric field (E-field), and the Hall conductivity is independent of Efield. In the case of a higher-order response regime, the presence of time-reversal symmetry rather allows for a new opportunity to encounter a novel Hall effect, namely a nonlinear Hall effect 3,[9][10][11] . Time-reversal symmetry in the absence of inversion symmetry enables a dipolar distribution of the Berry curvature, which leads to the nonzero integration of the Berry curvature gradient in momentum space, dubbed the Berry curvature dipole 3,4 . Consequently, a second-order nonlinear Hall voltage dominates the Hall effect, induced by the Berry curvature dipole, and the Hall voltage (conductivity) features a quadratic (linear) dependence on E-field.
Theoretically, it is possible that crystalline symmetries can further suppress the second-order nonlinear Hall effect with the help of time-reversal symmetry. Therefore, a higher-order nonlinear Hall effect can emerge 12-14 , which is yet to be addressed experimentally.
The Hall effect is responsible for not only electrical responses but also optical responses of electrons, which is referred to as the optical Hall effect 15 . The linear optical Hall response is widely used to measure magnetization of a system via magneto-optic Faraday effect, magnetooptic Kerr effect, or magnetic circular dichroism 16  In this work, we demonstrate, for the first time, the current-induced nonlinear optical Hall effect in multilayer WTe2 Weyl semimetal, which measures the analogue of the fourth-order nonlinear Hall effect under a time-reversal symmetric condition. The polarization rotation is measured at room temperature without application of magnetic field. In analogy with the magneto-optical Kerr rotation, which corresponds to a linear optical Hall effect, we call our observation as the nonlinear Kerr rotation. Our Kerr rotation should be distinguished from the well-known optical Kerr effect, which is 3 rd order nonlinear effect but is not the Hall effect.
The optical Kerr effect is driven by the diagonal part of the dielectric tensor, and its mechanism is related to the anharmonic potential of bound electrons. Whereas, our Kerr rotation is driven by the off-diagonal part of the dielectric tensor, and its mechanism is related to the Berry curvature (shown later). We give a precise fingerprint of the fourth-order nonlinearity by showing that the Kerr angle is proportional to the cube of the electric field: linear in the displacement field of charge current and quadratic in the proving field of the light. Our findings suggest that WTe2 can be used for the electrically controllable nonlinear-optic-medium. The For studying electro-optical phenomena in WTe2 devices, we adopt the polarization rotation microscopy in a reflection geometry with an application of oscillating charge current ⃗ (Fig. 2a). A pulsed laser with a linear polarization, the center wavelength 785 nm ( ∼ 1.6 eV/ℏ), and the pulse duration ~100 fs is applied to the devices. After reflection from the devices, polarization variation, e.g. rotation and ellipticity angles, is monitored (see Methods). Having established the fourth-order nonlinearity, we explore the anisotropic nature of the observed Kerr rotation. We perform a two-dimensional scanning of the Kerr rotation with different polarization directions of a linearly polarized light and displacement field as shown in Fig. 3. The Kerr rotation appears prominently throughout the sample area when the displacement field � �⃗ ∥ � and light field �⃗ ∥ � (Fig. 3b). When both � �⃗ and �⃗ are aligned along the -axis ( � �⃗ ∥ � and �⃗ ∥ � ), there still exists finite uniform signal, but the intensity is significantly weakened compared to the first case (Fig. 3c). More importantly, when � �⃗ ∥ �, it results in negligible signals, irrespective of the polarization direction as shown in Fig. 3e and We, then, quantitatively analyze the nonlinear Kerr rotation and make connection with the fourth-order nonlinear Hall conductivity. Under the action of the electric field �⃗ ( ) of the light incident along the sample -direction, the Kerr rotation with an initial polarization along -and -directions can be expressed as,

Current-induced Kerr rotation can be expressed as
where ( ≠ ), with subscript denotes the crystal axis, is the off-diagonal components of the dielectric tensor, which is related with the Hall conductivity as = As an example, when � �⃗ is applied along the sample -axis, and �⃗ is along the -axis (  (1) and (2), we obtain �  . 4). From the θ dependence of the Kerr rotation, we confirm the additional contribution from � (4) and � (4) . (To fit the experimental data, we use the Jones matrix formalism. See Supplementary Information S6). In addition, from the polar angle dependence, we identify contributions of � (4) and � (4) to the Kerr rotation. In particular, the magnitude of � (4) of > 2.7 × 10 −24 m 3 V −3 is much larger than those of � (4) and � (4) . As the fourth-order susceptibility is unexplored in other materials, to our best knowledge, it can be informative to convert the fourth-order susceptibility to the third-order susceptibility as . With Db of 6 × 10 3 V m −1 , we obtain � (3) of > 1.6 × 10 −20 m 2 V −2 , which is even larger than typical optical nonlinear systems, such as nanoparticle or polymer structures 38,39 . Therefore, our result demonstrates a new functionality, electrical control of the optical nonlinear process, with topological materials. Now, we provide a symmetry argument that explains the dominance of fourth-order nonlinear susceptibility (4) for the Kerr rotation of WTe2. Using the Boltzmann equation approaches, previous studies have shown that the fourth-order nonlinear susceptibility (4) can be expressed as 13,14 , where e is the elementary charge, ħ is the reduced Planck constant, FD is the Fermi-Dirac distribution, ≡ , � = + / , n is integer, ω is the angular frequency of light, is the relaxation time, and Ω is the Berry curvature with a pseudovector form of = For a mechanism other than the Berry curvature multipoles, the chirality of the Weyl point in Weyl semimetals can directly induce nonlinear optical processes, such as photocurrent or circular photogalvanic effect [21][22][23] . Interestingly, ref. 23 reported a photocurrent response that is forbidden by the symmetry of a material of TaIrTe4. Authors of ref. 23 interpreted the symmetryforbidden response as a result of combination of the built-in electric field and optical excitation.
Nonetheless, the chiral selection occurs with an optical transition from the lower part of the Weyl cone to the upper part, so it is mainly observed low photon energy. For connecting this effect to our work, further study is needed.
In summary, we observe the current-induced nonlinear optical Hall effect in WTe2 multilayers at room temperature. The optical Hall effect is revealed as the nonlinear Kerr rotation, and the fourth-order nonlinearity is confirmed by the linear dependence on the charge current and optical power. Our work shows that the nonlinear optical process can be controlled by a small electric bias, thereby, demonstrates a useful functionality for the electro-optic device employing topological materials. From the anisotropic behavior of the Hall response, we quantitatively determine various the fourth-order susceptibilities ( ). For the mechanism of (4) , we considered the Berry curvature hexapole, and it can explain the existence of � (4) , the largest susceptibility of our measurements. However, we found that some of our measurements, such as � (4) and � (4) , were forbidden by the symmetry of WTe2. The observation of the symmetry forbidden terms suggests that not only a stable phase but also other unknown phases participate during the nonlinear process. We expect that the fourth-order nonlinear optical Hall effect could be a useful tool to investigate the topological nature of Weyl semimetal.

Current-induced Kerr rotation microscopy:
A linearly polarized pulsed laser with the center wavelength of 785 nm and a pulse duration ~100 fs is used as the excitation source.
After reflection of the linearly polarized light from the sample, the polarization variation, e.g.
the rotation and the ellipticity angles, can be monitored. For the normal incidence geometry, the light polarization angle θ is defined with respect to the a-axis. The reflected laser beam from the sample is passing through a half-wave plate, Wollaston prism, and the balanced detector for monitoring polarization variation with/without quarter-wave plate before it.
Polarization variation can be precisely detected with a high sensitivity by using alternative bias induced modulation and a lock-in amplifier for increasing signal-to-noise ratio. The biasinduced Kerr angle can be expressed as, Kerr angle, is the electrical current density, and Δ � K is the variation of complex Kerr angle with an application of electrical bias (for details, see Supplementary Information S4).