Giant Berry curvature dipole density in a ferroelectric Weyl semimetal

The nonlinear Hall effect (NLHE) reflects Berry-curvature-related properties in non-centrosymmetric but time-reversal-symmetric materials. So far, the NLHE of the investigated systems remains a tiny effect due to the lack of Weyl point as magnetic monopoles in 2D systems or to the high carrier concentration in 3D systems. Here, we report large NLHE due to gigantic Berry curvature dipole density as generated by tilted Weyl cones near the Fermi level in a model ferroelectric Weyl semimetal In-doped Pb1−xSnxTe. By systematically lowering the carrier concentration down to ~1016 cm−3, the Berry curvature dipole density reaches values around 10−21 m3, 102–107 times higher than the previously reported ones. Furthermore, NLHE exhibits a power law of carrier concentration and follows the k−2 relation of the Berry curvature expression derived from the monopole. The present study establishes giant NLHE in a ferroelectric Weyl semimetal, promising for future applications such as current rectification.


INTRODUCTION
The Berry curvature (Ω) in electronic bands of topological quantum materials deeply affects their transport properties 1 . 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 materials 2 . 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 Orenstein 3 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 proposed 4 and experimentally verified 5,6 , and more recently observed in various materials without inversion symmetry, including artificially symmetry-broken or interface systems [7][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 R k f 0 ∂Ω ∂k with the Berry curvature Ω and the Fermi distribution function f 0 4 . D can also be expressed as the contribution from the states near the Fermi surface by the partial integration as D ¼ with the Fermi velocity 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 space [16][17][18][19][20] , which is described by the relation Ω k ð Þ ¼ ± k 2 k j j 3 . Due to the first k-derivative nature of the Berry curvature dipole 21 , 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 effect 4 . The NLHE is then promising for potential applications in wireless networks and energy harvesting [22][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, Indoped Pb 1−x Sn x Te 26,27 , 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 m 3 , which characterizes the strength of Berry curvature dipole irrespective of sample dimensions. By tracking the carrier concentration systematically down to ∼10 16 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.

Sample and characterization
The highly tunable topological materials system, (Pb 1−x Sn x ) 1−y In y Te, is engineered by adopting the Murakami's scheme for the WSM phase 18,28,29 , where a normal insulator (PbTe) and a topological crystalline insulator (SnTe) are alloyed to access the WSM phase steered by the intrinsic ferroelectric order 26,27,30 . 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 behaviors 27 . 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 ferroelectrics 27 . 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 model 31 are Fermi velocity and renormalized with a momentum cutoff Λ. Here, the number of Weyl node monopoles is N = 16, while dielectric constant ϵ ∼ 500 and hΛ $ 30 meV as derived from the optical conductivity data 27 . The T-linear dc conductivity is a typical feature of WSM 31 ; 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 v F (2 K) ∼ 1.2 × 10 5 m/s, which is consistent with the value (~1.7 × 10 5 m/s) we observe in the optical conductivity 27 .

NI TCI
Pb/Sn Te  1 Nonlinear Hall signals observed in a ferroelectric Weyl semimetal system of (Pb 1−x Sn x ) 1−y In y Te. a Schematic phase diagram of Indoped PbTe-SnTe; an intermediate Weyl seimimetal (WSM) phase appears between normal insulator (PbTe) and topological crystalline insulator (SnTe) with intrinsic ferroelectric order. Inset shows the distorted structure of WSM phase with polar axis along z axis. The fully mapped phase diagram with many data points can be found in ref. 26 b Temperature-dependent conductivity of the two prototypical samples, S33 and S36. Red linear, dashed line fits conductivity of S36 at temperature below 20 K. c In-plane Hall effects measured in S33 and S36 show vanishing and sizable signals, respectively. σ in-plane xz =p of S36 is adapted from ref. 27 d The Berry curvature dipole related by the mirror plane displays a certain structure of symmetry of sign, as indicated by the colors of d xy . e The nonlinear Hall effect (NLHE) measured for S33 and S36 shows vanishing and sizable signals, respectively. Inset shows the experimental configurations of current and Hall contacts. The error bars represent the standard deviation of the measured data.
This WSM phase is further confirmed by the in-plane Hall effect 27 , which is directly proportional to the weighted Berry curvature expressed as Ω / σ H==b xy =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 ð2ωÞ ¼ χ ijk E j E k , where the nonlinear susceptibility tensor is χ ijk ¼ Àϵ ilk Ã e 3 τ 2 h 2 D jl ; Berry curvature dipole is given by ∂ε Þ and the Berry curvature dipole density defined as d jl ¼ ∂Ω l ∂kj . Therefore, as shown in Fig. 1d, the d xy 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 d xy . For the WSM phase in (Pb 1−x Sn x ) 1−y In y Te, 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 points 27 , 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 E x with the frequency f = ω/2π will produce a 2ω Hall current along z axis by the χ zxx tensor, with the corresponding Berry curvature dipole D xy . 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 2ω zx . As shown in Fig. 1e, V 2ω zx 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 2ω zx 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 2ω zx and V 2ω xx in the transverse and longitudinal configuration are monitored simultaneously for sample S36, respectively. We can observe sizable signal of the NLHE V 2ω zx , while V 2ω xx remains almost vanishing as expected; the deviation on V 2ω zx 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, χ 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 level 27 . In order to quantify the NLHE 6 , we reformulate the relation  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 E 2ω ð Þ z E 2 x exhibits large enhancement at temperatures below 40 K. According to the Einstein's relation, the conductivity from Weyl cone is σ 0 ¼ e 2 vF 3π 2 h Ã k 2 F τ, and the Berry curvature dipole density is d xy ¼ and p. We note that the d xy 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 AHE 19 . As shown in Fig. 3a, the d xy calculated from experimental values exhibits a large value around 10 −21 m 3 at low temperatures, which is 10 2 -10 7 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 plot 32 , 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 Bilayer x Þ 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 F ¼ hk F v F $ 10 meV, where the Fermi wave vector k F is estimated from carrier concentration, which will induce large temperature-dependent carrier variation.
As show in Fig. 3b, The log-log plot of E 2ω with assuming a sole ∼k −2 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 σ in-plane xz =p $ p α . 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 (Pb 1−x Sn x ) 1−y In y Te 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 m 3 , 10 2 -10 7 times higher than the previously reported values, promising for the future applications.

Crystal growth and characterizations
Single-crystalline (Pb 1−x Sn x ) 1−y In y Te was grown by the conventional vertical Bridgman-Stockbarger technique 18 . 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 measurements 19 .

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).

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.