## Introduction

The Berry curvature (Ω) in electronic bands of topological quantum materials deeply affects their transport properties1. For instance, the Berry curvature contributes to the Hall conductivity via the momentum-space integral over all the occupied states, which results in the anomalous Hall effect (AHE) at zero magnetic field in magnetic materials2. Under the time reversal symmetry, however, we cannot observe finite AHE response in nonmagnetic materials at zero magnetic field even though finite Berry curvature exists on electronic band of a crystal lacking in inversion center. Moore and Orenstein3 realized that the cancellation of Berry curvature contribution, which is enforced by time reversal symmetry, can be removed by driving the system into non-equilibrium state, i.e., external electric field (E). In accord with this view, a type of Hall effect with no external magnetic field, namely the nonlinear Hall effect (NLHE), was proposed4 and experimentally verified5,6, and more recently observed in various materials without inversion symmetry, including artificially symmetry-broken or interface systems7,8,9,10,11,12,13,14,15. As opposed to the AHE described by the anomalous Hall conductivity, the NLHE is described by the second-order nonlinear susceptibility χ, which is in proportion to the Berry curvature dipole (D) defined as $${\int}_{{{\boldsymbol{k}}}} {f_0\frac{{\partial \Omega }}{{\partial {{{\boldsymbol{k}}}}}}}$$ with the Berry curvature Ω and the Fermi distribution function f04. D can also be expressed as the contribution from the states near the Fermi surface by the partial integration as $${{{\boldsymbol{D}}}} = {\int}_{{{\boldsymbol{k}}}} {{{{\mathrm{{\Omega}}}}}\left( {-\frac{{\partial f_0}}{{\partial {{{\boldsymbol{k}}}}}}} \right)} = {\int}_{{{\boldsymbol{k}}}} {{{{\mathrm{{\Omega}}}}}{{{\boldsymbol{v}}}}_F\left( { - \frac{{\partial f_0}}{{\partial \varepsilon }}} \right)}$$ with the Fermi velocity $${{{\boldsymbol{v}}}}_F$$ . This latter expression indicates that the nonlinear Hall current originates from the shift of the Fermi surface, while the former one stresses more its intrinsic nature. However, these two pictures are equivalent.

The most fundamental structure for emitting the Berry curvature flux is the monopole (Weyl node) in the k space16,17,18,19,20, which is described by the relation $$\Omega \left( k \right) = \pm \frac{{{{\boldsymbol{k}}}}}{{2\left| k \right|^3}}$$. Due to the first k-derivative nature of the Berry curvature dipole21, the NLHE is quite different from the AHE, e.g., in the chemical potential dependence, as exemplified by the analytical behaviors when the Fermi level approaches the monopoles. Therefore, the realization of a model Weyl semimetal (WSM) to show the well-defined NLHE is crucial for exploring the fundamental properties of the NLHE. The nonlinear susceptibility χ has two outcomes for the transverse current against the input ac electric field with angular frequency ω: one is the frequency doubling signal (2ω), another is the direct current signal representing the rectification effect4. The NLHE is then promising for potential applications in wireless networks and energy harvesting22,23,24,25. In analogy to the intrinsic magnetic order for the AHE, we then need a corresponding Landau order parameter as the control parameter of the NLHE. Therefore, it is highly demanding to realize a model system for the NLHE coupled strongly with the tunable order parameter that serves as a control knob to turn on/off the NLHE and tune the strength of the inversion symmetry breaking.

In this work, we report on the NLHE of an ideal WSM, In-doped Pb1−xSnxTe26,27, which shows the strong coupling with ferroelectric order. We observe sizable NLHE signal appearing along the polar axis, and can be turned on/off by tuning the ferroelectric order through temperature ramping and chemical composition variation. Furthermore, the observed NLHE signal supports a largest value of Berry curvature dipole density reaching as high as 10−21 m3, which characterizes the strength of Berry curvature dipole irrespective of sample dimensions. By tracking the carrier concentration systematically down to 1016 cm−3, we obtain the scaling law between Berry curvature dipole (as well as the effective Berry curvature) and the carrier density. The resultant exponents are found to be consistent with the results calculated based on the simplest model, namely the isolated Weyl monopoles with no other overlapping trivial band dispersions.

## Results

### Sample and characterization

The highly tunable topological materials system, (Pb1−xSnx)1−yInyTe, is engineered by adopting the Murakami’s scheme for the WSM phase18,28,29, where a normal insulator (PbTe) and a topological crystalline insulator (SnTe) are alloyed to access the WSM phase steered by the intrinsic ferroelectric order26,27,30. As shown in Fig. 1a, the schematic phase diagram exhibits a hot composition-zone containing the WSM phase with ferroelectric order, while samples out of the hot zone exhibit non-WSM behaviors27. The Indium doping (y) is adopted to reduce the bulk carrier concentration, and subsequently to foster the ferroelectric order via reducing the free carrier screening. The ferroelectric transition was evidenced by the temperature-dependent second harmonic generation (SHG) probing the electric polarization as well as by the appearance of a soft phonon mode characteristic of the displacive-type ferroelectrics27. The Indium doping becomes more efficient when composition Sn (x) becomes less rich, which makes the regime of WSM tilted with ferroelectric order appearing at Indium (y) concentration in the range of 0.01–0.05, as shown in Fig. 1a. In this study, we use the two prototypical samples, namely S33 (x = 0.48, y = 0.05; non-WSM without ferroelectric order) and S36 (x = 0.42, y = 0.04; WSM with ferroelectric order), to explore the NLHE. Figure 1b shows the temperature-dependent dc conductivity of S33 and S36, respectively. The conductivity of S36 exhibits temperature (T)-linear dependence below 20 K where the phonon scattering can be ignored, and Coulomb effects dominate. This is consistent with the proposed formula of the conductivity based on the simplest Weyl model31: $$\sigma _{dc} = \frac{{e^2}}{h} \ast \frac{{k_BT}}{{\hbar v_F\left( T \right)}}\frac{{0.45}}{{\alpha _T^2{{{\mathrm{ln}}}}\alpha _T^{ - 1}}}$$, where $$v_F( T) = v_F( {\frac{{\alpha _0}}{{\alpha _T}}})^{\frac{2}{{N + 2}}}$$ and $$\alpha _T = \alpha _0\left[ {1 + \frac{{\left( {N + 2} \right)\alpha _0}}{{3\pi }}\ln \left( {\frac{{\hbar {{{\mathrm{{\Lambda}}}}}}}{{{{{k}}}_{{{\mathrm{B}}}}T}}} \right)} \right]^{ - 1}$$ are Fermi velocity and renormalized fine structure $$\left( {\alpha _0 = \frac{{e^2}}{\hbar } \ast \frac{1}{{4\pi {\it{\epsilon }}v_F}}} \right)$$ with a momentum cutoff $${{{\mathrm{{\Lambda}}}}}$$. Here, the number of Weyl node monopoles is N = 16, while dielectric constant $${\it{\epsilon }}$$ 500 and $$\hbar {\Lambda}\sim 30\;{{{\mathrm{meV}}}}$$ as derived from the optical conductivity data27. The T-linear dc conductivity is a typical feature of WSM31; which was not accessible in previously reported WSMs due to the overwhelming contribution from trivial bands crossing near the Fermi level. As shown in Fig. 1b, the T-linear fitting to the conductivity of S36 extrapolates to zero conductivity at 0 K, although there is slight deviation due perhaps to the impurity scattering and ferroelectric hysteresis at lower temperatures. The fitting procedure gives Fermi velocity vF (2 K) 1.2 × 105 m/s, which is consistent with the value (~ 1.7 × 105 m/s) we observe in the optical conductivity27. This WSM phase is further confirmed by the in-plane Hall effect27, which is directly proportional to the weighted Berry curvature expressed as $${{{\mathrm{{\Omega}}}}} \propto \sigma _{xy}^{H//{{{\boldsymbol{b}}}}}/p$$. Here the magnetic field is applied along b (in-plane; we adopt the coordinate relation, x(a), y(−c), z(b) in this paper) direction, and p is the hole-type carrier concentration. The sample S36 exhibits a large weighted Berry curvature in contrast with a vanishing value observed in S33, as shown in Fig. 1c.

### Nonlinear Hall effect

Provided the WSM phase with the Berry curvature around the monopoles, it is of great interest to explore the second order effect of the Berry curvature. Figure 1d introduces the simplest model for the Berry curvature dipole to produce the NLHE. The NLHE describes the generation of frequency-doubling current as: $$j^{(2\omega )} = \chi _{ijk}E_jE_k$$, where the nonlinear susceptibility tensor is $$\chi _{ijk} = - {\it{\epsilon }}_{ilk} \ast \frac{{e^3\tau }}{{2\hbar ^2}}D_{jl}$$; Berry curvature dipole is given by $$D_{jl} = {\int}_k {f_0\frac{{\partial {\it{\Omega }}_l}}{{\partial k_j}}} = {\int}_{{{\boldsymbol{k}}}} {{{{\mathrm{{\Omega}}}}}{{{\boldsymbol{v}}}}_F( { - \frac{{\partial f_0}}{{\partial \varepsilon }}})}$$ and the Berry curvature dipole density defined as $$d_{jl} = \frac{{\partial {\it{\Omega }}_l}}{{\partial k_j}}$$. Therefore, as shown in Fig. 1d, the dxy exhibits opposite signs in different orthants, and hence the Weyl cone needs tilting to accumulate finite values. Furthermore, the Weyl monopoles, related by the mirror or time reversal symmetry, contribute additively to the total dxy. For the WSM phase in (Pb1−xSnx)1−yInyTe, the polar axis is along z, and the Weyl monopoles, associated with slightly tilted Weyl cones due to arbitrary locations of monopoles away from high symmetric k points27, are related by the remaining mirror planes and time reversal symmetry, which allows us to use a single Weyl cone model to analyze the observed NLHE signals.

By considering the symmetry constraint (see Supplementary Note 2 for the detailed argument), the ac electric field Ex with the frequency f = ω/2π will produce a 2ω Hall current along z axis by the χzxx tensor, with the corresponding Berry curvature dipole Dxy. As shown in the inset of Fig. 1e, the current with f = 13.333 Hz is fed along x direction, and the 2ω signal of the NLHE along z axis is denoted as $$V_{zx}^{2\omega }$$. As shown in Fig. 1e, $$V_{zx}^{2\omega }$$ shows sizable signals and depends on the square of the excitation current in the WSM S36 with ferroelectric distortion, while the same measurement produces almost a negligible value in the non-WSM S33. This controlled experiment shows the on/off behaviors of the NLHE directly coupled with the presence/absence of ferroelectricity. We also reverse the current and Hall contacts simultaneously to find $$V_{zx}^{2\omega }$$ reversed in sign but with the same magnitude, which confirms the nonlinear origin of the second order. Furthermore, we also check another configuration with current fed along z axis and Hall voltage contacts along x axis, which should produce no NLHE according to the symmetry analysis (see Supplementary Note 2). As shown in Supplementary Fig. 3, we find the NLHE is much smaller than that along z axis. Therefore, we conclude that the observed 2ω Hall signal originates from the NLHE produced by the Weyl monopoles.

### Temperature dependence of the nonlinear Hall effect

We now investigate the temperature dependence of NLHE, and study the variation of Berry curvature dipole with carrier concentration. As shown in Fig. 2a, $$V_{zx}^{2\omega }$$ and $$V_{xx}^{2\omega }$$ in the transverse and longitudinal configuration are monitored simultaneously for sample S36, respectively. We can observe sizable signal of the NLHE $$V_{zx}^{2\omega }$$, while $$V_{xx}^{2\omega }$$ remains almost vanishing as expected; the deviation on $$V_{zx}^{2\omega }$$ as discerned at higher current regime is caused perhaps by thermoelectric signals induced by current-heating effect. Furthermore, no frequency dependence of the NLHE is observed at least in the range of f = 3–300 Hz, as shown in Supplementary Fig. 2, confirming negligible thermoelectric effect as a fictious source of nonlinear response. Figures 2a–d show that the NLHE gradually decreases when temperature is raised from 3 to 20 K, while the polar distortion is almost kept constant in this temperature range according to the optical second harmonic generation result in ref. 27 At temperatures higher than 40 K, the NLHE signal (see Supplementary Fig. 4) completely disappears due perhaps to the increase of carrier concentration or change of band tilting, which needs further study to elucidate the origin.

### Analysis of the Berry curvature dipole density and scaling law

The nonlinear susceptibility tensor, $$\chi _{ijk}$$, contains the Berry curvature dipole (D) and transport lifetime (τ). Therefore, the carrier concentration and dc conductivity determine the NLHE and hence may cause complicated temperature dependence of the NLHE when the target system has a complicated band structure like conventional WSMs with trivial pockets around the Fermi level. In this context, the target system here is ideal for investigating the purely Weyl monopole based NLHE effect because of the simple band structure composed of the Weyl cones alone near the Fermi level27. In order to quantify the NLHE6, we reformulate the relation as $$\chi _{zxx} = \frac{{j^{\left( {2\omega } \right)}}}{{E_x^2}} = \frac{{E_z^{\left( {2\omega } \right)}}}{{E_x^2}} \ast \sigma _0$$, where $$\sigma _0$$ is the dc conductivity. We can see that the experimental quantity $$\frac{{E_z^{\left( {2\omega } \right)}}}{{E_x^2}}$$ is independent of the relaxation time τand hence the intrinsic quantity for the band structure. More explicitly, $$\frac{{E_z^{\left( {2\omega } \right)}}}{{E_x^2}} = \frac{{e^3\tau }}{{2\hbar ^2}}\frac{{{{{\mathrm{D}}}}_{xy}}}{{\sigma _{xx}}} = \frac{e}{{2\hbar ^2}}\frac{{ < {{{\mathrm{{\Omega}}}}}_yv_x > }}{{ < v_x^2 > }}$$where < > represents the average with the weight $$- \frac{{\partial f_0}}{{\partial \varepsilon }}$$, which is reduced to the average over the Fermi surface at zero temperature. For the Weyl fermion, the tilting is essential for the nonzero NLHE. When the chemical potential μ is near the Weyl point, there occurs contributions both from the electrons and holes at finite temperature. In sharp contrast to the linear Hall effect, the contributions to the nonlinear Hall effect from electrons and holes have the same sign, and hence we can regard the carrier density $$p$$ as that for the shifted μ away from the Weyl point, or the thermally induced carrier density of both electrons and holes when μ is at the Weyl point. In Fig. 3a, the quantity $$\frac{{E_z^{\left( {2\omega } \right)}}}{{E_x^2}}$$ exhibits large enhancement at temperatures below 40 K. According to the Einstein’s relation, the conductivity from Weyl cone is $$\sigma _0 = \frac{{e^2v_F}}{{3\pi ^2\hbar }} \ast k_F^2\tau$$, and the Berry curvature dipole density is $$d_{xy} = \frac{{D_{xy}}}{p} = \frac{{E_z^{\left( {2\omega } \right)}}}{{E_x^2}} \ast \frac{{2\hbar v_F}}{e}\left( {3\pi ^2p} \right)^{ - \frac{1}{3}}.$$Then, dxy can be obtained from experimentally observed quantities, $$\frac{{E_z^{\left( {2\omega } \right)}}}{{E_x^2}}$$, vF19 and p. We note that the dxy has no unit dependence on dimension, is a universal physical quantity for representing the strength of Berry curvature dipole effect like the averaged Berry curvature we adopted for AHE19. As shown in Fig. 3a, the dxy calculated from experimental values exhibits a large value around 10−21 m3 at low temperatures, which is 102–107 times higher than the reported values for 2D or 3D systems; Table 1 summarizes to compare the reported values on different materials.

With keeping such a temperature-dependent NLHE in mind, we identify the feature of the NLHE of the present system. In the case of AHE, a useful scheme for unraveling the intrinsic Berry curvature contribution is the so-called Onoda-Nagaosa plot32, which scales Hall conductivity with longitudinal conductivity. This scaling law was simulated by using a Rashba model with an assumption that the Fermi energy is constant, thus the conductivity as the x-axis variable is dependent only on the impurity scattering time τ. However, topological semimetals usually exhibit very low carrier concentration and strongly temperature-dependent carrier dynamics, which make it difficult to adopt this scheme to study the intrinsic contribution from Berry curvature or its dipoles. Here, to reveal the dominant contribution of intrinsic Berry curvature dipole, we focus on a scaling scheme for Berry curvature dipole $$( { \propto \frac{{E_z^{\left( {2\omega } \right)}}}{{E_x^2}}})$$ versus carrier concentration (p), which can be identified in such a system composed of pure monopoles near the Fermi level as the present one. The Fermi level of S36 is close to the monopoles with the energy distance of about $$E_{{{\mathrm{F}}}} = \hbar k_Fv_F\sim 10\;{{{\mathrm{meV}}}}$$, where the Fermi wave vector kF is estimated from carrier concentration, which will induce large temperature-dependent carrier variation. As show in Fig. 3b, The log–log plot of $$\frac{{E_z^{\left( {2\omega } \right)}}}{{E_x^2}}$$ versus p produces an exponent of β −0.8 in the form of $$\frac{{E_z^{\left( {2\omega } \right)}}}{{E_x^2}}\sim p^\beta$$. This is consistent with the formula $$\frac{{E_z^{\left( {2\omega } \right)}}}{{E_x^2}} = \frac{{\chi _{zxx}}}{{\sigma _0}}\sim < {{{\mathrm{{\Omega}}}}} > \sim \frac{1}{{k_F^2}}\sim p^{ - \frac{2}{3}}$$ with assuming a sole k2 Berry curvature contribution from the monopole. This scaling law confirms that the observed NLHE originates from intrinsic Berry curvature dipole around the Weyl monopole, which is the sole Berry curvature flux source in the present system. The scaling of the weighted Berry curvature inferred from the in-plane Hall effect is also plotted against carrier concentration p in Fig. 3b, which produces an exponent α −0.4 in the form of $$\sigma _{xz}^{{\rm{in}} {\mbox{-}} {\rm{plane}}}/p\sim p^\alpha$$. This exponent is also roughly in accord with the value (α = −2/3) expected from the expression of the monopole, which again confirms that the Berry curvature is dominated by the occupied state of the Weyl cones alone.

## Discussion

In conclusion, we have established a model ferroelectric Weyl semimetal system (Pb1−xSnx)1−yInyTe for exploring the NLHE generated by the tilted monopoles. The effective Berry curvature dipole derived from the experimentally observed nonlinear Hall voltage follows a scaling law with carrier concentration, which is consistent with the simplest form of the Berry curvature dipole expected for the Weyl monopoles. Furthermore, the observed Berry curvature dipole density, which characterizes the strength of the NLHE, manifests the largest value as high as 10−21 m3, 102–107 times higher than the previously reported values, promising for the future applications.

## Methods

### Crystal growth and characterizations

Single-crystalline (Pb1−xSnx)1−yInyTe was grown by the conventional vertical Bridgman–Stockbarger technique18. Laue and energy dispersion X-ray (EDX) characterizations were performed to determine the crystal axes and specific compositions. The polar axis is determined by the second harmonic generation measurements19.

### Transport measurements

Nonlinear electrical transport measurements were carried out in a Quantum Design physical property measurement system (PPMS) with a multifunctional probe, which is modified to accommodate the coaxial cables. The four-probe method by using the ultrasound indium welding is used for preparing electrodes. For the nonlinear measurements, 2f signals are monitored by the lock-in technique with π/2 phase shift, compared with that of excitation current (f = 13.333 Hz).