Geometrical Nonlinear Hall Effect Induced by Lorentz Force
Junjie Yao Yizhou Liu Wenhui Duan









State Key Laboratory of Low Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing, 100084, China


yizhouliu@tongji.edu.cn
School of Physics Science and Engineering, Tongji University, Shanghai 200092, China
Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, Tongji University, Shanghai 200092, China


State Key Laboratory of Low Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing, 100084, China
Collaborative Innovation Center of Quantum Matter, Beijing 100084, China
Institute for Advanced Study, Tsinghua University, Beijing 100084, China







             

§ ABSTRACT 
The recently discovered nonlinear Hall (NLH) effect arises either without external magnetic field (type-I) or with an in-plane magnetic field (type-II). In this work we propose a new type of geometrical nonlinear Hall effect with an out-of-plane magnetic field (type-III) induced by the combination of Lorentz force and anomalous electronic velocity. The type-III NLH effect is proportional to the more refined structures of Bloch wave functions, i.e., the dipole moment of square of Berry curvature, thus becoming prominent near the band crossings or anticrossings. Our effective model analysis and first-principles calculations show that gate-tuned MnBi_2Te_4 thin film under uniaxial strain is an ideal platform to observe this effect. Especially, giant unidirectional magnetoresistance can occur in this material, based on which an efficient electrical transistor device prototype can be built. Finally a symmetry analysis indicates that type-III NLH effect has unique symmetry properties stemming from Berry curvature square dipole, which is different from other previously reported NLH effects and can exist in a wider class of magnetic crystals. Our study offers new paradigms for nonlinear electronics.



                               
Geometrical Nonlinear Hall Effect Induced by Lorentz Force
    Wenhui Duan
    September 9, 2024
==========================================================





§ INTRODUCTION
 The Hall effects, as one of the most long-history and yet paradigmatic phenomena in condensed matter physics, have been extensively studied due to their underlying rich physics <cit.>. Interestingly, the Hall effects can have two very different physical origins: they can be induced by the Lorentz force in the external magnetic field, or result from the electronic band geometry and topology <cit.>. Prominent examples of the latter include the quantum Hall effect in a strong magnetic field <cit.> and its anomalous version without a magnetic field <cit.>, which have provided valuable insights into the nontrivial electronic structures related to the momentum space textures (i.e., curvature and metric) of Bloch wave functions, and have triggered exotic applications in many other systems <cit.>. The former, at lower magnetic fields, is classical and does not rely on Bloch wave function textures <cit.>.





On the other hand, the recently discovered nonlinear anomalous Hall (NLAH) effect has stimulated the interest of nonlinear electronic transport and optoelectronic studies. This effect connects the nonlinear response coefficients with some geometrical or topological properties of Bloch wave functions, and has promising applications, including second harmonic generation, radio-frequency ac-dc rectification, and terahertz detection, among others <cit.>. The NLAH effect refers to the Hall current j^H∝ E^2 in response to the driving electrical field E without external magnetic field as shown in Fig. <ref> (a), which we dub type-I nonlinear Hall (NLH) effect and is related to the dipole moment of Berry curvature or quantum metric. More recently, another type of nonlinear planar Hall effect was proposed with j^H∝ E^2 B where the Hall current j^H, driving electrical field E, and magnetic field B lie within the same plane <cit.> as shown in Fig. <ref>(b), which we call type-II NLH effect and corresponds to the Berry connection polarizability. In the standard Hall geometry, as shown in Fig. <ref> (c), the ordinary linear Hall effect induced by Lorentz force usually dominates, with the Hall current j^H∝τ^2 EB (where τ is the relaxation time), and resultant  ordinary Hall conductivity depends only on the carrier density, independent of the electronic wave function properties <cit.> (see also Appendix <ref>). However, the nonlinear properties of the Lorentz force-induced Hall effect have not been explored so far. This raises a straightforward question: does a Lorentz force-induced NLH effect exist, and could this NLH effect further connect the measurable quantities with any refined intrinsic properties of Bloch electrons, beyond just the carrier density like the ordinary Hall effect?

In this paper we show that the Lorentz force together with the Berry-curvature-induced anomalous velocity <cit.> can lead to a new type of geometrical NLH effect (type-III) with j^H∝τ E^2 B, which is in lower power of τ compared to the ordinary Hall effect and is related to the more refined geometrical properties of Bloch wave functions, offering a straightforward method to investigate this more intricate geometrical properties in materials. Different from the type-I NLH effect induced by either Berry curvature dipole (BCD) <cit.> or quantum metric dipole <cit.>, the proposed type-III NLH effect here results from the Berry curvature square dipole which dominates in topological materials with large Berry curvature and can exist in a broader class of magnetic point groups. Based on effective model analysis and density functional theory (DFT) calculations we find sizable type-III NLH conductivity in gate-tuned MnBi_2Te_4 double septuble layers (SLs) under moderate uniaxial strain. Moreover, we also find giant unidirectional magnetoresistance (UMR) effect in this material system without strain, which can be utilized to build an electronic transistor device prototype. Finally, a symmetry analysis is carried out to identify all magnetic point groups that allow the existence of our NLH and UMR effects. The unique symmetry properties of Berry curvature square dipole can help us find some systems that allow only type-III NLH while forbid type-I NLH effects mentioned above. Our findings pave the way for further investigation on this new type of nonlinear Hall effect and possible electronic device applications.



§ THEORY AND ANALYSIS



 §.§ Theory of type-III NLH effect
 To understand the origin of type-III NLH effect, we start from the well-known anomalous Hall current of solids <cit.>:

    j^H = -e^2/ħ∫dk/(2π)^d E×Ω f,

where E is the external driving electrical field; Ω = i ⟨∇_k u_k | × | ∇_k u_k⟩ is the Berry curvature with | u_k⟩ being the periodic part of the Bloch wave function; f is the electronic occupation number and d refers to the spatial dimension. According to Eq. (<ref>), the linear anomalous Hall conductivity is totally determined by the electronic Berry curvature at thermodynamic equilibrium, which is finally determined by the unperturbed electronic wave function of occupied states. Thus in order to get nonlinear Hall effects, one must consider the field-induced perturbation effect on either the electronic wave function |u_k⟩ or the distribution function f. The NLH effects corresponding to the former one is an intrinsic material property which can be expressed in terms of Berry connection polarizability and quantum metric tensors <cit.>.

On the other hand the perturbation on the distribution function f can be determined by Boltzmann equation under relaxation time approximation:

    - f - f_0/τ = k̇·∇_k f + ṙ·∇_r f + ∂_t f

with f_0 = [ e^(ε_k - ε_F)/k_BT + 1 ]^-1 being the equilibrium Fermi-Dirac distribution function. Combined with the semiclassical equation of motion of electrons <cit.>, the nonequilibrium contribution to the nonlinear Hall current can be derived as (see details in Appendix <ref>):

    j̃^NLH = -e^2/ħ∫dk/(2π)^d E×Ωδ f, 
    δ f= f - f_0 = -τ( F_E ·ṽ + F_L ·v_A ) ∂ f_0/∂ε_k + O(τ^2).

Here F_E = -e E is the electrical force, and ṽ = v+v_M is the total group velocity, with v_M= ∇_k (-m_k·B)/ ħ, and  v = ∇_kε_k/ħ the ordinary group velocity of electron. The ordinary group velocity part in the first term on the right-hand side of Eq. (<ref>) corresponds to the Berry curvature dipole contribution to the nonliner anomalous Hall effect <cit.>, and has been extensively studied in inversion-symmetry-breaking nonmagnetic materials <cit.>.

The second term, on the other hand, is the Lorentz-force-induced contribution which is seldom discussed before and will be the focus of this paper hereafter. Here, F_L = -e v×B refers to the Lorentz force which is perpendicular to the group velocity. Therefore, the Lorentz force does not induce energy shift in non-topological materials with zero Berry curvature. However, in topological materials the velocity of electrons acquires an anomalous term v_A = (-e/ħ) E×Ω induced by the Berry curvature effects <cit.>. The F_L ·v_A τ can be viewed as the energy shift induced by the Lorentz force and anomalous velocity which is generally nonzero and generates the Fermi surface shift shown in Fig. <ref> (d). It should be noted that the magnetic moment-related part in the first term on the right hand side of Eq. (<ref>) is of the same order O(τ E^2B) as the Lorentz force-induced one but exhibits relatively small magnitude <cit.>. Therefore, we will focus only on the Lorentz-force part.

Equations (<ref>) and (<ref>) demonstrate a new type of NLH effect induced by the combination of Lorentz force and the anomalous velocity. Given that F_L ∝B and v_A ∝E, the NLH current is proportional to τ E^2B and it can be expressed in a compact form as j^NLH_α = ∑_β,γ,λσ^NLH_αβγλ E_β E_γ B_λ with σ^NLH_αβγλ being the NLH conductivity whose expression is given by

    σ^NLH_αβγλ = e^4τ/ħ^2∫dk/(2π)^d ∑_κϵ_αβκΩ_κ ( v_γΩ_λ - δ_γλ∑_μv_μΩ_μ ) ∂ f_0/∂ε_k,

where Greek indexes represents Cartesian coordinates, ϵ_αβκ is the Levi-Civita tensor, and δ_γλ is the Kronecker symbol. Recall that the group velocity v is odd under both inversion symmetry (𝒫) and time-reversal symmetry (𝒯) while the Berry curvature Ω is even under 𝒫 but odd under 𝒯 so σ^NLH_αβγλ exists only when 𝒫 and 𝒯 are simultaneously broken.



In addition to the intrinsic contribution from Berry curvature, disorder scatterings, such as skew scattering and side jumps, also play a role in linear Hall conductivity <cit.>. These scatterings are naturally expected to influence the nonlinear Hall (NLH) conductivity as well <cit.>. The impact of disorder scattering is highly dependent on carrier density: it is minimal when the Fermi energy is near the band edge but becomes significant as carrier density increases <cit.>. Conversely, Eq. (<ref>) demonstrates that the NLH conductivity proposed here is proportional to the square of the Berry curvature on the Fermi surface. This indicates its prominence in small-gap topological materials when the Fermi level is close to the band edges. The stark contrast between the contributions from Berry curvature and disorder scatterings helps to distinguish the mechanisms behind NLH conductivity through electronic doping effects.



 §.§ Berry curvature square dipole
 For 2D materials lying within xy plane, in order that the Lorentz force can take effects the magnetic field must have z component. This is reflected by fact that the NLH conductivity σ^NLH_αβγλ is nonzero only when λ=z, which is dramatically different from the nonlinear planar Hall effect <cit.>. Moreover, the second term on the right hand side of Eq. (<ref>) vanishes for 2D materials because v⊥Ω. Thus the NLH conductivity can be rewritten as:

    σ^NLH_yxxz = -σ^NLH_xyxz = e^4τ/ħ^3∫dk/(2π)^2  (∂_k_xΩ^2_z) f_0, 
    σ^NLH_xyyz = -σ^NLH_yxyz = -e^4τ/ħ^3∫dk/(2π)^2  (∂_k_yΩ^2_z) f_0,

while other components are zero. The above expressions show that the NLH conductivity is proportional to the dipole moment of square of Berry curvature over occupied states, which is defined as

    𝒟^(n)_α = ∫dk/(2π)^2  (∂_k_αΩ^n_z) f_0,

with n=2. Because 𝒟^(n)_α behaves like a vector within the 2D plane, the NLH conductivity shows a peculiar angular dependence: for applied electric field E = E(cosθ, sinθ, 0), the NLH conductivity is determined by σ^NLH(θ) = (e^4τ/ħ^3) ( 𝒟^(2)_x cosθ + 𝒟^(2)_y sinθ ). It should be noted that 𝒟^(2)_α is different from the ordinary BCD i.e. 𝒟^(1)_α first proposed in Ref. <cit.>. Under vertical mirror symmetry, 𝒟^(1)_α behaves like a pseudo vector which should be perpendicular to the mirror plane, while 𝒟^(2)_α behaves like a real vector and should be parallel to the mirror plane. Moreover, 𝒟^(2)_α is expected to become more prominent than 𝒟^(1)_α near the band crossings and anticrossings due to the large Berry curvature.






§ RESULTS AND DISCUSSIONS



 §.§ Effective Model analysis
 We take the 2D massive Dirac model as an example to demonstrate the behavior of NLH conductivity here. Without loss of generality we consider σ^NLH_yxxz here as an example. A nonzero σ^NLH_yxxz requires the breaking of mirror symmetry M_x. The minimal effective model can be expressed as,

    H_k= ħ v_t k_x + ħ v_0(k_xσ_x + k_yσ_y) + mσ_z,

with v_0 being the group velocity, m being the mass term, and v_t being the tilting parameter to break 𝒫, 𝒯 and M_x symmetries. The band dispersion and Berry curvature are  ε_sk = ħ v_t k_x + s √(ħ^2 v_0^2 k^2+ m^2), and Ω_z = -smħ^2 v^2_0/2(ħ^2 v^2_0 k^2 + m^2)^3/2 (s= ± represents the upper or lower band), respectively, with a band gap of 2|m|. For small tilting, we can derive that σ^NLH_yxxz(ε_F) = e^4τ m^2 v^2_0 v_t/8π|ε_F|^5( 1- 3m^2/|ε_F|^2) + O(v^2_t) with |ε_F| ≥ m to ensure nonzero density of states <cit.>. The maximum σ^NLH_yxxz occurs at the band edges |ε_F|=m, and the value is proportional to m^-3 which is prominent in small-gap materials.


Figure <ref> shows the numerical results for the effective model. For untilted model, the band resolved σ^NLH_yxxz exhibits opposite values between k and -k which gives vanishing σ^NLH_yxxz for any ε_F so finite tilting is necessary to get nonzero σ^NLH_yxxz [Figs. <ref>(a)-(b)]. The calculated σ^NLH_yxxz increases with increasing v_t and the maximum values occurs near the band edge [Fig. <ref>(c)], which is consistent with the analytical result. Figures <ref>(d)-(e) show the calculated maximum σ^NLH_yxxz with varying v_t and m favoring large tilting and small band gap.






 §.§ Candidate Material
 According to the effective model analysis, 𝒫- and 𝒯- breaking materials with small band gap (i.e., near the topological phase transition) favor large NLH conductivity. MnBi_2Te_4 is a recently discovered layered topological antiferromagnet which attracts extensive research of interest because of its unique axion dynamics and versitile topological phase transitions <cit.>. Remarkably, recent studies have shown the gate-tunable topological properties of MnBi_2Te_4 thin film <cit.> with multiple Dirac Fermions near the Fermi energy under suitable vertical electric gating field. Figure <ref>(a) shows the crystal structure and Brillouin zone of MnBi_2Te_4 double septuple layers (SLs). A vertical electric field E_⊥ breaks the 𝒫𝒯 symmetry thus giving rise to finite Berry curvature. Without the vertical electric field, MnBi_2Te_4 with double SLs is a topologically trivial insulator with spin degeneracy protected by 𝒫𝒯 [Fig. <ref>(b)]. Applying the vertical electric field will split the energy bands and reduce the band gap <cit.>
. As increasing E_⊥, the band gap closes at about a critical field E_⊥ = E_c = 0.022 V/Åwith a tilted Dirac cone at the Fermi level [Fig. <ref>(c)]. Due to the 3-fold rotational symmetry C_3z, there are three tilted Dirac cones whose tilting directions are related by C_3z. Therefore, the net NLH conductivity vanishes [Fig. <ref>(d)]. By applying an external uniaxial strain the C_3z is broken and a nonzero NLH conductivity arises. Figure <ref>(e) shows the calculated σ^NLH_yxxz as function of Fermi energy ε_F with a large peak which favors further experimental measurement.





 §.§ Unidirectional magnetoresistance
 In the presence of external magnetic field, in addition to the NLH effect discussed above, there is another intriguing transport phenomenon that the longitudinal conductivity σ changes with the direction of current, which is called electrical magnetrochiral anisotropy or unidirectional magnetoresistance (UMR) effect, and has the same order O(E^2B) with NLH. The UMR effect in gated MnBi_2Te_4 double SLs can be described by current- and magnetic-field- dependent conductivity σ_αβ(j,B) = σ^0_αβ( 1 + ∑_γ,λΛ_αβγλ j_γ B_λ) with σ^0_αβ being the ordinary Drude conductivity and Λ_αβγλ being an intrinsic material property. Based on semiclassical equation of motion, the UMR is dominated by the Berry curvature on the Fermi surface, which is confirmed by the analytical results in a 3D Weyl model <cit.>. Figure <ref>(a) shows the calculated Λ_xxxz under perpendicular electrical field E_⊥ = ± E_c. The calculated UMR shows sharp peaks of about 2×10^4 m A^-1 T^-1 around ε_F = 0 and its sign reverses with the direction of E_⊥, which means the magnitude of electrical conductivity changes dramatically for a sample under current j ≈ 1 μA/cm and B_z=1 T. Such a nonlinear conductivity can be used to realize a electrical transistor device prototype under a small gating field V_0 which induces E_⊥ = ± E_c [Figs. <ref>(b)-(c)]. Besides, we find that NLH effect can exhibit sizable signals up to moderate temperature, however, UMR effect survives only in the low-teperature regime <cit.>.





 §.§ Symmetry analysis
 Now we analyze the symmetry properties of NLH and UMR effects. The UMR generally requires inversion symmetry breaking and can exist in all non-centrosymmetric magnetic point groups. On the other hand, the NLH effect is more complicated. Type-III NLH effect has a different physical origin from the BCD induced NLAH and the intrinsic nonlinear Hall effect (INHE), both of which belong to the type-I NLH effect, thus has different symmetry requirements. By careful analysis, we give the list of magnetic point groups that allow the existence or absence of type-III NLH effect, along with that of BCD and INHE, as shown in Table <ref> which is obtained based on the magnetic tensor symmetry module implemented in Bilbao Crystallographic Server <cit.>. It should be noted that BCD is fully forbidden by the coexistence of a mirror symmetry (which forbids the diagonal parts of BCD) and a rotation symmetry (which forbids the off-diagonal parts of BCD), and the case is similar for INHE which origins from quantum metric dipole, while type-III NLH effect is related to the Berry curvature square dipole which can at least have nonzero diagonal elements in the presence of a mirror symmetry. Obviously, we can find some magnetic groups where our type-III NLH exists but BCD induced NLAH and INHE are forbidden (the first row of Table <ref>). There are also some magnetic groups that allow BCD but forbid type-III NLH effect, which all have time reversal symmetry (the fourth row of Table <ref>). Additionally, from Table <ref>, we can clearly see that the type-III NLH effect can exist in a wider range among all the magnetic point groups (69 of 122 magnetic point groups allow type-III NLH) than BCD and INHE (53 of 122 magnetic point groups allow BCD and INHE). This broader range may benefit further experimental investigation and verification.




§ SUMMARY
 We propose a theory of a new type of nonlinear Hall effect, which stems from the combination of the Lorentz force and anomalous velocity. Based on first-principles calculations, we show MnBi_2Te_4 double SLs as an ideal candidate with significant NLH conductivity. Interestingly, a giant UMR effect is also predicted in this material system with full electrical tunability, which may be useful for developing a new generation of high-performance electrically switchable transistor without PN junction. Our symmetry analysis further shows that this NLH effect can exist in a wider range of magnetic point groups compared to previously reported mechanisms, which may facilitate further investigations into these effects. Our finding highlights the important role of the Lorentz force in exploring the more refined electronic structure and topology of materials, which may have been previously overlooked.



§ ACKNOWLEDGEMENTS

This work was supported by the Innovation Program for Quantum Science and Technology (Grant No. 2023ZD0300500), the Basic Science Center Project of NSFC (Grant No. 51788104), and the Beijing Advanced Innovation Center for Future Chip. Y.L. is sponsored by the Shanghai Pujiang Program (Grant No. 23PJ1413000), NSFC (Grant No. 12404279) and the Fundamental Research Funds for the Central Universities.









§ DERIVATION OF ORDINARY HALL (OH) CONDUCTIVITY 


In this section we derive the expression of ordinary Hall (OH) conductivity for a nearly free electronic gas model that supports a parabolic band. For a nearly free electron gas (without Berry curvature or orbital magnetic moment), the semiclassical equation of motion is

    ħk̇ =    -eE -eṙ×B, 
    ṙ =   v = 1/ħ∇_kε_k.

The Boltzmann transport equation is

    ( ∂_t + k̇·∇_k + ṙ·∇_r) f = - f - f_0/τ,

i.e.,

    [ 1 + τ( ∂_t + k̇·∇_k + ṙ·∇_r) ] f = f_0.

Because τ is usually small, the above Eq. (<ref>) can be formally solved as

    f =   [ 1 + τ( ∂_t + k̇·∇_k + ṙ·∇_r) ]^-1 f_0 
    
            =   ∑^+∞_l=0[ - τ( ∂_t + k̇·∇_k + ṙ·∇_r) ]^l f_0 
    
            =   ∑^+∞_l=0 f_l,

where f_l ∝τ^l is the l-th order perturbation term for the distribution function. For the steady and uniform distribution, i.e., ∂_t f = ∇_r f =0, the expression of f_l can be simplified as

    f_l = ( -τk̇·∇_k)^l f_0.


Now we can derive the expression of OH conductivity based on Eqs. (<ref>) and (<ref>). Substitute Eq. (<ref>) into (<ref>), we get the first order term as

    f_1 =   eτ/ħ ( E + v×B) ·∇_k f_0 
    
            =   eτ/ħ ( E + v×B ) ·ħv∂ f_0/∂ε_k
    
            =    eτE·v∂ f_0/∂ε_k.

From Eq. (<ref>), we can see that Lorentz force F_L =- ev×B does not enter into the first-order term because it is perpendicular to the group velocity v. As a result, it does not induce a Fermi surface shift. The second order term is

    f_2 =    -τk̇·∇_k f_1 
    
            =   eτ/ħ (E + v×B) ·∇_k[ e τE·v∂ f_0/∂ε_k] 
    
            =   e^2τ^2/ħ ( E + v×B ) 
       ·[ ∇_k (v·E) ∂ f_0/∂ε_k + (E·v) ∂^2 f_0/∂ε^2_kħv].

We focus on the Lorentz-force induced terms in Eq. (<ref>) which will lead to the OH conductivity:

    f_OH   = -τ^2 e^2/ħ (v×B) ·∇_k (v·E) ∂ f_0/∂ε_k
       =τ^2 e^2/ħ [ ∇_k (v·E) ×B] ·v∂ f_0/∂ε_k
       = τ^2 e^2 v·[ 1/m^*E×B] ∂ f_0/∂ε_k,

where the dispersion relation of nearly free electron gas ε_k = ħ^2 k^2/2m^* and the inverse effective mass (1/m^*)_αβ = 1/ħ^2∂ε_k/∂ k_α k_β have been used. Thus the ordinary Hall current is expressed by

    j^OH =    -∫dk/(2π)^d  ev f_OH
    
            =    - e^3τ^2/m^*∫dk/(2π)^d vv·( E×B) ∂ f_0/∂ϵ_k.

For isotropic 3D electron gas, the OH effect can be expressed as j^OH = σ^OH (E×B) with the ordinary Hall conductivity calculated as

    σ^OH =    -e^3τ^2/m^*∫dk/(2π)^3 v^2/3∂ f_0/∂ε_k
    ≈   e^3 τ^2/3m^* (2π)^3∫ dk ħ^2 k^2/(m^*)^2δ(ε_F - ħ^2 k^2/2m^*) 
    
            =   2e^3τ^2/3(m^*)^2 (2π)^3∫^+∞_0 dk  4π k^2 k^2δ(k_F - k)/2k_F
    
            =   4π e^3 τ^2 /3(m^*)^2 (2π)^3 k^3_F,

where k_F = √(2m^* ε_F)/ħ is the Fermi wave vector, and ∂ f_0/∂ε_k≈ - δ(ε_F - ε_k) has been used in the above derivation. On the other hand, the carrier density is

    n_F =   ∫dk/(2π)^3 f_0 = ∫^k_F_0 dk 4π k^2/(2π)^3 = k^3_F/6π^2.

By substitute Eq. (<ref>) into (<ref>), we get the familiar expression of OH conductivity as

    σ^OH = e^3τ^2n_F/(m^*)^2.

Two key points about the ordinary Hall effect:

    
  * It is of second order in τ;
    
  * It relies only on the band dispersion relation ε_k and is independent of the Bloch wave function |u_k⟩.





§ DERIVATION OF TYPE-III NONLINEAR HALL (NLH) EFFECT 


In this section, we derive the expression of NLH effect based on semiclassical equation of motion of electrons including Berry curvature and orbital magnetic moments:

    ħk̇ =    -e E - e ṙ×B, 
    ṙ =   1/ħ∇_kε_M - k̇×Ω,

where ε_M = ε_k - m·B (m refers to the orbital magnetic moment), and Ω is Berry curvature. Equation (<ref>) can be solved as:

    D k̇ = -e/ħE - e/ħṽ×B - e^2/ħ^2 (B·E) Ω, 
    
        D ṙ = ṽ + e/ħE×Ω + e/ħ (ṽ·Ω) B,

with ṽ = 1/ħ∇_kε_M being the modified group velocity, and D = ( 1 + e/ħΩ·B)^-1.

We expand the distribution function f in a power series of relaxation time τ as: f = f_0 + f_1 + f_2 + ⋯ with f_n ∝τ^n and f_0 = [ e^(ε_k - ε_F)/k_BT + 1 ]^-1 being the equilibrium distribution function. According to the Boltzmann equation, i.e., Eq. (<ref>) of the main text, we have

    f_n =    - τk̇·∇_k f_n-1,

for spatially uniform E and B. Substituting Eq. (<ref>) into Eq. (<ref>) we can derive the first-order term f_1 as:

    f_1    = τ/D[ e/ħE + e/ħṽ×B + e^2/ħ^2 (B·E) Ω] ·∇_k f_0 
       = τ/D[ e/ħE + e/ħṽ×B + e^2/ħ^2 (B·E) Ω] · (∇_kε_M) ∂ f_0/∂ε_M
       ≈τ/D[ eE·ṽ + e ṽ×B·ṽ + e^2/ħ (B·E) (Ω·ṽ) ] ∂ f_0/∂ε_k
       ≈τ/D[ eE·ṽ + e^2/ħ (B·E) (Ω·v)  ] ∂ f_0/∂ε_k,

where v = 1/ħ∇_kε_k is the group velocity. On the third line of Eq. (<ref>) we have used the approximation ∂ f_0/∂ε_M≈∂ f_0/∂ε_k and on the last line we have substituted some ṽ terms by v_k by omitting a O(B^2) term in f_1. For small magnetic field D ≈ 1 - e/ħΩ·B. Thus the expression of f_1 up to linear order of B can be simplified as

    f_1 =   τ[ eE·ṽ( 1 - e/ħΩ·B) + e^2/ħ (B·E) (Ω·v) ] ∂ f_0/∂ε_k
    
            =   τ[ eE·ṽ - e^2/ħ (E·v) (Ω·B) + e^2/ħ (B·E) (Ω·v) ] ∂ f_0/∂ε_k
    
            =   τ[ eE·ṽ - e^2/ħ (v×B) · (E×Ω) ] ∂ f_0/∂ε_k
    
            =    -τ( F_E ·ṽ + F_L ·v_A ) ∂ f_0/∂ε_k,

with F_E = -eE, v_A = (-e/ħ) E×Ω, and F_L = -e v×B.















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