Disorder-induced nonlinear Hall effect with time-reversal symmetry

The nonlinear Hall effect has opened the door towards deeper understanding of topological states of matter. Disorder plays indispensable roles in various linear Hall effects, such as the localization in the quantized Hall effects and the extrinsic mechanisms of the anomalous, spin, and valley Hall effects. Unlike in the linear Hall effects, disorder enters the nonlinear Hall effect even in the leading order. Here, we derive the formulas of the nonlinear Hall conductivity in the presence of disorder scattering. We apply the formulas to calculate the nonlinear Hall response of the tilted 2D Dirac model, which is the symmetry-allowed minimal model for the nonlinear Hall effect and can serve as a building block in realistic band structures. More importantly, we construct the general scaling law of the nonlinear Hall effect, which may help in experiments to distinguish disorder-induced contributions to the nonlinear Hall effect in the future.

T he Hall effects refer to a transverse voltage in response to a current applied in a sample of metal or semiconductor. The family of the classical and quantized Hall effects is one of the mainstreams of modern condensed matter physics, leading to the full spectrum of the search on the topological states of matter and many practical applications 1,2 . All previous Hall effects are in the linear-response regime, that is, the transverse voltage is linearly proportional to the driving current, and a measurable Hall voltage requires that time-reversal symmetry is broken by magnetic fields or magnetism [1][2][3][4] . The recently discovered nonlinear Hall effect [5][6][7][8][9][10][11][12][13] does not need time-reversal symmetry breaking but inversion symmetry breaking, significantly different from the known linear Hall effects (Fig. 1). The linear Hall effects can be understood in terms of the Berry curvature 14 , which describes bending of a parameter space (real space, momentum space, any vector fields) 15 . This geometric description is of the same significance as the curved spacetime in the general theory of relativity. The nonlinear Hall effect depends on the higher-order properties of the Berry curvature, thus not only can bring our knowledge to the next level but also may help device applications. More importantly, by adjusting the measurements to the nonlinear regime, unconventional transport phenomena can be explored in a great number of emergent materials in which discrete and crystal symmetries are broken.
The disorder effects have been a large part of the research on the linear Hall effects, such as the localization in the quantized Hall effects 16,17 , the extrinsic mechanisms of the anomalous 3 , spin 18 , and valley 19 Hall effect, etc. The debate on the origin of the anomalous Hall effect lasted for one century, until recently the mechanisms are summarized in terms of intrinsic (disorder-free) and extrinsic (disorder-induced) contributions 3 . The quantitative agreement between theories and experiments shows that the disorder-induced contribution is comparably important 20,21 . In the nonlinear Hall effect, disorder is more important, because the effect always requires that the Fermi energy crosses an energy band, while on the Fermi surface the disorder scattering is inevitable and enters the nonlinear Hall effect even in the leading order. This is quite different from the disorder-free leading order in the linear Hall effects. How disorder contributes to the nonlinear Hall signal in a specific form remains unknown and is the focus of the investigations at this stage.
In this work, we use the Boltzmann formalism to derive the formulas of the nonlinear Hall conductivity in the presence of disorder scattering. The formulas can be applied to different models to calculate the nonlinear Hall responses. We apply the formula to the 2D tilted massive Dirac model. The model is a symmetry-allowed minimal model for the nonlinear Hall effect and can be used to understand the nonlinear Hall signals in realistic band structures 12 . Depending on roles of disorder scattering, we follow the convention to classify the nonlinear Hall conductivity into the intrinsic, side-jump, and skew-scattering contributions. The latter two are new findings to the framework of the nonlinear Hall effect and comparably important. The competition between the three contributions can induce a sign change in the nonlinear Hall signal. More importantly, we present the scaling laws of the nonlinear Hall effect, which help to identify distinct contributions and explain the temperature and thickness dependence in the experiments in the future.

Results
General formulas. The nonlinear Hall effect is measured as zeroand double-frequency transverse electric currents driven by a low-frequency ac longitudinal electric field JðEÞ ¼ Àe P l _ r l f l in the absence of a magnetic field, where the ac electric field EðtÞ ¼ RefEe iωt g with the amplitude vector E and frequency ω. Here −e is the electron charge, l ¼ ðη; kÞ labels a state in band η with wave vector k and f l is the corresponding distribution function. The current up to the second-order of the ac electric field can be found as J a ¼ RefJ respectively, where fa; b; cg 2 fx; y; zg. Table 1 summarizes our main results for the anomalous Hall response tensor σ ab and the double-frequency nonlinear Hall response tensor χ abc (see Methods). We have assumed that ωτ ( 1, because ω is about tens of Hertz and the relaxation time τ is about picoseconds in experiments. This low-frequency limit is one of the differences from the nonlinear optics. The disorder-induced zero-frequency response ξ abc is identical with the double-frequency response χ abc in the ωτ ( 1 limit. Away from the ωτ ( 1 limit, the doubleand zero-frequency nonlinear Hall conductivities have different frequency dependence, thus are different in general. For a complete description, we list the ω-dependent full expressions with and without time-reversal symmetry (Supplementary Note 3), which would be helpful for understanding the recently proposed high-frequency rectification 22 and gyrotropic Hall effects 23 . According to how disorder works, the formulas are classified in terms of intrinsic (in), side-jump (sj), skew-scattering (sk) contributions. The formulas in Table 1 can be applied to different models to calculate the nonlinear Hall responses. . σ A is the anomalous Hall conductivity, which is always anti-symmetric 3 . M represents the magnetization. σ P is the planar Hall conductivity. Δσ σ k À σ ? , where σ k and σ ? refer to the longitudinal conductivities along the two principal axes. θ is the angle between the driving current and the k principal axis (the dashed lines). σ N is the nonlinear Hall conductivity, which is proportional to the magnitude of the driving electric field E.
where ðk x ; k y Þ are the wave vectors, ðσ x ; σ y ; σ z Þ are the Pauli matrices, t, v, and m are model parameters. t tilts the Dirac cone along the x direction. The time reversal of the model contributes equally to the Berry dipole, so it is enough to study this model only. For the disorder part, we assume a δ-correlated spin independent random potentialV imp ðrÞ ¼ To have analytic expressions with intuitive insight, we assume that t ( v and the relaxation time τ is k-independent, i.e., where n i is the impurity density (Supplementary Note 4). As functions of the Fermi energy ε F , we obtain the analytic expressions for the intrinsic side-jump and skew-scattering response functions up to the linear order in t. χ xyy ¼ 0 for each contribution, as required by mirror reflection symmetry k y $ Àk y . According to the above analytic expressions, the side-jump (χ sj ) and intrinsic skew-scattering (χ sk;1 ) contributions are of the same order with the intrinsic one (χ in ). The extrinsic skew-scattering (χ sk;2 ) contribution is controlled by the relative scattering strength of the non-Gaussian scattering V 3 1 . The factor ε 2 F À m 2 in all the contributions secures that the nonlinear Hall conductivity vanishes at the band edge. It is interesting to note that the sidejump contribution dominates near the bottom of the band, which is consistent with the result of a recent work 24 . At higher ε F , the skew-scattering becomes the strongest contribution, which is similar to the behaviors in the anomalous Hall effect. All contributions vanish as ε F ! 1. These behaviors can be seen in Fig. 2e.
The zero-frequency nonlinear Hall response was not addressed experimentally. In the ωτ ( 1 limit, ξ yxx ¼ χ yxx . According to symmetry, ξ xyy ¼ 0. In the dc limit (ω ¼ 0), the electric field becomes time independent E a ðtÞ ¼ E a , and the nonlinear Hall response becomes a dc current  (2)], an x-direction electric field can generate a dc nonlinear Hall current along the y direction. As a result, the measured Hall conductivity will be proportional to the electric field In contrast, if the electric field is along the y direction, there is no such a Hall signal because χ xyy ¼ 0, as required by the ydirection mirror reflection symmetry. This indicates that the dc nonlinear Hall signal σ N xy has one-fold angular dependence 11,12 . This dc Hall signal can exist in the presence of time-reversal symmetry, which has been observed in the nonmagnetic Weyl-Kondo semimetal Ce 3 Bi 4 Pd 3 25 .
Scaling law of the nonlinear Hall effect. It is of fundamental importance to distinguish the different contributions to the nonlinear Hall signal in experiments. For the anomalous Hall effect, distinguishing different contributions is based on the scaling law of the transverse Hall signal to the longitudinal signal 3,20,21,26 . For the nonlinear Hall effect, a scaling law can be constructed as well. We adopt the quantity V N y =ðV L x Þ 2 ¼ ξ yxx ρ xx or χ yxx ρ xx as the experimental scaling variable 11,12 , where V N y and V L x refer to the nonlinear Hall (zero-or double-frequency) and linear longitudinal voltage, respectively. To measure the nonzero χ yxx , the driving electric current is applied along the x direction and the nonlinear Hall voltage is measured along the y direction. An advantage of this variable is that the intrinsic and side-jump parts become disorder independent.
To account for multiple sources of scattering 21,26 , we consider the scaling law of nonlinear Hall effect in a general manner. For simplicity, we assume no correlation between different scattering sources, thus each source contributes to the total resistivity independently, as dictated by Matthiessen's rule ρ xx ¼ P i ρ i 27 , where ρ i is the contribution of the ith type of disorder scattering to the longitudinal resistivity. According to Table 1, the general scaling law of the nonlinear Hall effect can be obtained as Side-jump (velocity) σ sj;1 ab ¼ À P l v sj a g b l χ sj;1 abc ¼ À Intrinsic skew-scattering σ sk;1 ab ¼ À Extrinsic skew-scattering σ sk;2 ab ¼ À We refer to the leading-order of the nonlinear Hall conductivity as the intrinsic contribution, but it depends on the disorder scattering, quite different from the disorder-free intrinsic Hall conductivity. The side-jump and skew-scattering contributions are due to the coordinates shift and antisymmetric scattering, respectively. Here ε abc is the anti-symmetric tensor, we define ∂ a ∂=∂k a , ∂ ′ a ∂=∂k ′ a , and g a l τ l ∂ a f ð0Þ l . τ l is the general relaxation time and f ð0Þ l is the Fermi distribution. The Berry curvature 3, 15 Ω a l = −2ε abc P l′≠l Imhlj∂ bĤ jl′i hl′j∂ cĤ jli/ðε l À ε l′ Þ 2 , where jli is the eigenvector. The side-jump velocity v sj a ¼ P l′ ϖ sy ll′ δr a l′l andM ab l P l′ ðM ac ll′ ÀM ac l′l Þδðε l À ε l′ Þ, where ϖ sy ll′ is the symmetric scattering rate, the coordinates shift 28 , δr a ll′ = ihlj∂ a jli − ihl′j∂ ′ a jl′i − ð∂ a þ ∂ ′ a Þ argðV ll′ Þ with V ll′ hljV imp jl′i and M ab ll′ ð2π= hÞ∂ a ðτ l jT ll′ j 2 δr b ll′ Þ. ϖ g ll′ and ϖ ng ll′ refer to the Gaussian and non-Gaussian antisymmetric scattering rate, U a ll′ τ l v a l À τ l′ v a l′ andŨ ab Here the disorder-independent coefficients are for the intrinsic (C in ), side-jump (C sj i ), intrinsic skew-scattering (C sk;1 i ), and extrinsic skew-scattering (C sk;2 ij ) contributions, respectively. S stands for static disorder scattering sources 21 . To use Eq. (7), one needs to specify scattering sources. As an example, we consider two major scattering sources 21 , one static (i ¼ 0) and one dynamic (i ¼ 1), then the scaling law becomes with four scaling parameters ð9Þ C 1;2;3;4 can be extracted from experiments 20,21,26 . Here ρ xx0 is the residual resistivity due to static impurities at zero temperature and ρ xxT ρ xx À ρ xx0 is due to dynamic disorders (e.g., phonons) at finite temperature.
In the zero-temperature limit (T ! 0), we can approximate that ρ xxT ' 0 and ρ xx ' ρ xx0 ¼ σ À1 xx0 , then the scaling law becomes V N y =ðV L x Þ 2 ' C 1 σ xx0 þ C 2 , which indicates a linear scaling behavior as shown in Fig. 3a. Fitting the experimental data using this relation, the extrinsic skew-scattering coefficient C sk;2 can be experimentally extracted from the total nonlinear Hall conductivity (e.g., by using multi-step samples 20,21,26 ). Furthermore, at finite temperatures, it is more convenient to rewrite the scaling law into In this case, the proper scaling variable becomes V N y =ðV L x Þ 2 À C 1 σ À1 xx0 σ 2 xx , which is a parabolic function of σ À1 xx0 σ xx thus indicates a scaling behavior shown in Fig. 3b. By fitting the  Fig. 3 Scaling law of the nonlinear Hall effect. a Step 1. At zero temperature, fitting C 1 and C 2 with the data of V N y =ðV L x Þ 2 and σ xx0 for samples of different disorder strength (e.g., by changing the thickness 20,21 ). Insert is the schematic of the WTe 2 multi-step sample. b Step 2. At finite temperatures, for a given sample of known σ xx0 , fitting C 3 and C 4 with the data of σ xx at different temperatures. experimental data with the parabolic function, one can in principle extract the information of the rest scaling parameters, as shown in Fig. 3. Equation (10) can be reorganized as In the anomalous Hall effect, the second term on the right has been argued to be negligible in both the hightemperature limit (σ xx0 ) σ xx ) and the low-temperature limit (σ xx0 ' σ xx ) 20 . This linear scaling behavior has been observed in thin films of WTe 2 11 . Nevertheless, Eq. (10) shows that the linear scaling behavior with σ 2 xx may become invalid in the highconductivity regime 21,26 . In the nonmagnetic Weyl-Kondo semimetal Ce 3 Bi 4 Pd 3 , a linear scaling behavior of σ N xy is observed as a function of σ xx 25 . The scaling law of voltages in Eq. (10) can be written into the scaling law of the nonlinear Hall conductivity for a fixed electric field. According to the conductivity scaling law, the observed linear behavior 25 indicates the dominance of the scaling parameter C 4 . According to Eq. (9), C 4 is contributed mainly by the intrinsic mechanism and the dynamical scattering processes (e.g., Fig. 2b, c).

Methods
Boltzmann formalism in the nonlinear regime. In the Boltzmann formalism, the distribution function f l can be found from the standard Boltzmann equation 27 , which reads in the spatially uniform case. Here I el ff l g represents the elastic disorder scattering by static defects or impurities. The elastic disorder scattering can be decomposed as the intrinsic, side-jump, and skew-scattering parts (Supplementary Note 1) I el ff l g ¼ I in el ff l g þ I sj el ff l g þ I sk el ff l g: ð13Þ The intrinsic part is contributed by symmetric scatterings, in which incoming and outgoing states are reversible in a scattering event. The side-jump part is resulting from the coordinates shift during scattering processes. The skewscattering part is contributed by anti-symmetric scatterings, in which exchanging the incoming and outgoing states yields a minus sign. Specifically, I in el ff l g ¼ À P l′ ϖ sy ll′ ðf l À f l′ Þ, I sj el ff l g ¼ ÀeE Á P l′ O ll′ ðf l À f l′ Þ, I sk el ff l g ¼ À P l′ ϖ as l′l ðf l þ f l′ Þ, where ϖ sy ll′ and ϖ as ll′ represent the symmetric and antisymmetric parts of the scattering rate ϖ ll′ ¼ ð2π= hÞjT ll′ j 2 δðε l À ε l′ Þ with T ll′ representing the T-matrix 27 . O ll′ ð2π= hÞjT ll′ j 2 δr ll′ ∂ ∂ε l δðε l À ε l′ Þ, where the coordinates shift δr ll′ is defined in Table 1. The expression of _ r l and _ k can be found from the semiclassical equations of motion 15,28 where v l ¼ ∂ε l = h∂k is the group velocity, Ω l is the Berry curvature 15 , and v sj l is the side-jump velocity 28 (see Table 1). To solve the Boltzmann equations up to the second order of E, we adopt the relaxation time approximation 27 for the intrinsic scattering parts I in el ff l g ¼ ðf is the Fermi distribution function and τ l represents the relaxation time. Usually, in good metal regime, τ l is treated as a constant that can be determined by experiments. For systems with large anisotropy, τ l can have a significant angular dependence 29,30 . With the above equations, the current up to the second-order responses to the ac electric field can be obtained.
Tilted 2D massive Dirac model with disorder. We use the tilted 2D massive Dirac model in Eq. (2) to calculate the nonlinear Hall conductivity in Fig. 2. The model describes two energy bands (denoted as ±) with the band dispersions ε ± k ¼ tk x ± ½v 2 k 2 þ m 2 1=2 , where k 2 k 2 x þ k 2 y . In the x-y plane, the Berry curvature behaves like a pseudoscalar, with only the z component Ω z ± k ¼ Çmv 2 =½2ðv 2 k 2 þ m 2 Þ 3=2 .
To consider the disorder effect, we expanded the scattering rate up to the fourth order in the disorder strength as ϖ ll′ ¼ ϖ

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.

Code availability
The code that is deemed central to the conclusions is available from the corresponding author upon reasonable request.