Electric-field induced nonequilibrium orbital magnetization and resulting nonlinear charge and valley Hall effect

Abstract

We provide deeper insights into the nonlinear transports in strained monolayer graphene and find that both nonlinear valley and nonlinear charge Hall effects can be interpreted with the orbital magnetic moment (OMM). Since strain induced anisotropic velocities and band-warping terms break the inversion and rotation symmetry, the nonlinear valley and charge Hall effect emerges. We demonstrate that the intrinsic OMM, originating from Berry curvature linearly corrected by electric field, is valley-contrasting which contributes to the nonlinear valley Hall current, and the shift OMM, originating from Fermi distribution function linearly corrected by electric field, is valley-independent which contributes to the nonlinear Berry-curvature-dipole (BCD) Hall current. Thus, we reveal that the nonlinear BCD Hall current and nonlinear valley current essentially have the same physics and the dependence of valley index of the orbital magnetic moment determines which nonlinear Hall effect emerges. Physically, we give an interpretation of two-step process: One electric field E_x induces an orbital magnetization and then the other electric field E_x generates the anomalous Hall effect under the orbital magnetization. These results establish a microscopic connection between orbital magnetization and nonlinear Hall responses. Furthermore, the strong dependence of OMM on strain provides a route to strain-engineered control of the nonlinear Hall transports.

Graphical abstract

Keywords

nonlinear Hall effect / Berry curvature dipole

1 Introduction

Conventional Hall effect and anomalous Hall effect [1, 2] require time-reversal symmetry broken by a magnetic field or magnetism. Recently, nonlinear anomalous Hall effect was proposed as a second-order response to an electric field, which originates from interesting geometric quantities, i.e., Berry curvature dipole (BCD) [3, 4] and quantum metric dipole (QMD) [5–8]. This nonlinear anomalous Hall effect does not require time-reversal symmetry breaking but inversion symmetry breaking. The nonlinear anomalous Hall effect has been theoretically predicted in various noncentrosymmetric materials, such as two-dimensional transition-metal dichalcognides (TMD) [3, 9, 10] and three-dimensional topological materials [11–13], and has been experimentally observed, such as in Weyl semimetal WTe₂ [4] or TaIrTe₄ [14] and strained monolayer WSe₂ [15].

The nonvanishing nonlinear Hall effect in conventional materials attributes to a common trait: Strong spin-orbit coupling and the presence of low-energy Dirac quasiparticles forming titled Dirac cones. Recently, it was reported that sizable BCD can be caused by warping of the fermi surface in strained graphene [16–18], in which the spin-orbit coupling and titled Dirac cones are completely absent. The nonzero BCD in strained graphene is ascribed to the higher order warping of Fermi surface and gives rise to nonlinear Hall effect. Recently, it was found that the nonlinear anomalous Nernst effect also appears in strained graphene as a result of trigonal warping of the Fermi surface [19]. Thus, monolayer graphene or TMD provides an exceptional platform to modulate and investigate the BCD through strain. In these time-reversal invariant two-dimensional strained materials, the nonlinear Hall effect usually stems from the BCD but not the QMD which needs breaking time-reversal symmetry [20].

Above scenario is suitable for the nonlinear response with respect to the charge degrees of freedom. Except for the charge, graphene or TMD usually has valley degrees of freedom. Valleys are the degenerate energy extrema of the electronic bands in momentum space and become well-defined degrees of freedom when they are well separated in momentum space with negligible intervalley scattering. The valley degree of freedom has attracted significant attention, giving birth to the field of valleytronics [21–23]. Intriguingly, very recently, Das et al. [24] proposed a nonlinear valley Hall effect, which can be induced by QMD even in time-reversal symmetry, in spite of vanishing nonlinear charge Hall effect. As extending nonlinear response from charge degrees of freedom to valley, on one hand, the requirement for symmetry changes, and on the other hand, there arises a fundamental question: Whether we can give common interpretation about the nonlinear charge and nonlinear valley Hall effect. In linear valley Hall effect [25] and nonlinear valley Hall effect [24], both are ascribed to orbital magnetic moment (OMM) [26, 27]. We expect the nonlinear charge Hall effect also can be interpreted with OMM. Thus, we can understand the microscopic origin of both the nonlinear charge and nonlinear valley Hall effect in terms of the OMM [28–31].

In this work, we give a common interpretation about the nonlinear charge and nonlinear valley Hall effect. We reveal that both the charge/valley nonlinear responses with respect to the quadratical electric field {\mathit{E}}^2 can be understood with a two-step process: The first step is generation of a nonequilibrium OMM by one electric field \mathit{E} and then under the OMM the Hall effect is formed by another electric field \mathit{E}. While the previous works attributed the nonlinear Hall effect under the time-reversal symmetry to the BCD mechanism, we provide deeper insights with respect to the nonequilibrium OMM. Thus, we bridge between seemingly different effects — nonlinear charge and nonlinear valley hall effects, and unify them into a single physical picture based on different types of nonequilibrium OMM. Specifically, we find that the nonequilibrium OMM induced by shift of Fermi surface is valley-independent, which leads to the BCD induced nonlinear charge Hall current [26, 32–36]. In contrast, the nonequilibrium orbital magnetization induced by electric field corrected Berry curvature or QMD is valley-contrasting, which leads to QMD-induced nonlinear valley Hall effect. Therefore, the nonlinear charge current and nonlinear valley current essentially have the same physics and the dependence of valley index of the OMM determines which nonlinear Hall effect emerges. We investigate them by employing the strained graphene under uniaxial deformation. Furthermore, it is found that the nonlinear charge/valley Hall current can be distinguished from each other since they appears along orthogonality direction. We also in detail discuss the influence of various parameters, such as strain, mass gap, and chemical potential.

2 Model and theory

We consider electrons hopping on a honeycomb lattice, whose dynamics are governed by the nearest-neighbor tight-binding Hamiltonian

(1)H={\displaystyle \sum }_{\mathit{R},\delta }t(\mathit{R},\delta )c_A^{†}(\mathit{R})c_B(\mathit{R}+\delta )+H.C.,

where \mathit{R} denotes a position on the Bravais lattice and \delta is the vector connecting this site to its nearest neighbors. The operators c_A(\mathit{R}) and c_B(\mathit{R}) correspond to the field operators on the A and B sublattices, respectively. We only need to consider three independent hopping parameters: t_1=t\left({\delta }_1\right), t_2=t\left({\delta }_2\right), and t_3=t\left({\delta }_3\right). When one chooses x-axis along zigzag direction, the relative positions of the three nearest-neighbor B-sublattice atoms with respect to a given A-sublattice atom are given by {\delta }_1=\left(0,a/\sqrt{3}\right), {\delta }_2=\left(a/2,-a/2\sqrt{3}\right), and {\delta }_3=\left(-a/2,-a/2\sqrt{3}\right). Here a denotes the lattice constant.

By transforming Eq. (1) into the momentum space, one can write the two-band Hamiltonian as H=\sum _{\mathit{q}}{c}^{†}(\mathit{q})H(\mathit{q})c(\mathit{q}), where c(\mathit{q})=[c_A(\mathit{q}),c_B(\mathit{q}){]}^{\mathrm{T}} with

(2)H\left(\mathit{q}\right)=\left(\begin{array}{cc}{\displaystyle \frac{\mathrm{\Delta }}{2}}& -{\displaystyle \sum _n{t}_n{\mathrm{e}}^{\mathrm{i}\mathit{q}\cdot {\delta }_n}}\\ -{\displaystyle \sum _n{t}_n^{\ast }{\mathrm{e}}^{-\mathrm{i}\mathit{q}\cdot {\delta }_n}}& -{\displaystyle \frac{\mathrm{\Delta }}{2}}\end{array}\right).

Here, \mathrm{\Delta } is a staggered potential between the two honeycomb sublattices, which can be introduced by placing the graphene sheet on a substrate, for example, lattice-matched h-BN [37, 38]. This staggered chemical potential breaks inversion symmetry and allows for a nonvanishing Berry curvature.

In the absence of strain, due to the existence of a threefold rotation C_{3z}, after integration of the momentum, the BCD vanishes. When a specific strain is applied to the system, the nearest-neighbor hopping amplitudes are modulated by the strain [17, 39] as t_n=t_0\left(1-\beta \mathrm{\Delta }{u}_n\right), where \beta is a material-dependent parameter, taking the value \beta \simeq 2 for graphene. The quantity \mathrm{\Delta }{u}_n can be expressed in terms of the strain tensor {\epsilon }_{i,j} as \mathrm{\Delta }{u}_n=3{\delta }_n^i{\epsilon }_{i,j}{\delta }_n^j/a^2, where {\delta }_n^i denotes the i-th component of the n-th nearest-neighbor vector. Throughout this work, we use the equilibrium carbon–carbon bond length a=1.42 Å as the unit of length.

By expanding Eq. (2) around the two inequivalent valleys {\mathit{K}}_{\pm }=\left(\pm 4\pi /3a\mp \frac{v_F{A}_x}{v_x},0\right), where we define the Fermi velocity as v_F=\sqrt{3}{t}_{0}a/2, and the term including A_x=\sqrt{3}\beta ({\epsilon }_{xx}-{\epsilon }_{yy})/2a is the shift of two valleys caused by the strain, we can obtain a low-energy effective Hamiltonian as [17]

(3)\begin{array}{rl}{H}_{warped}=& \xi v_x{k}_x{\sigma }_x+v_y{k}_y{\sigma }_y\\ & +\left({\lambda }_1{k}_y^2-{\lambda }_2{k}_x^2\right){\sigma }_x+2\xi {\lambda }_2{k}_x{k}_y{\sigma }_y+\frac{\mathrm{\Delta }}{2}{\sigma }_z.\end{array}

Here, \xi =\pm denotes the valley index, \sigma refers to the Pauli matrices acting on the sublattice degree of freedom. Under the strain, the velocities become anisotropic in the form of v_x=v_F\left[1-\beta \left(3{\epsilon }_{xx}+{\epsilon }_{yy}\right)/4\right] and v_y=v_F[1-\beta \left({\epsilon }_{xx}+3{\epsilon }_{yy}\right)/4]. As retaining terms up to the second order in momentum, the band-warping effects are presented with the warping coefficients given by {\lambda }_1={\lambda }_0\left[1-\beta \left(5{\epsilon }_{yy}-{\epsilon }_{xx}\right)/4\right] and {\lambda }_2={\lambda }_0[1-\beta (3{\epsilon }_{xx}+{\epsilon }_{yy})/4] with {\lambda }_0=t_0{a}^2/8. Anisotropic dispersion v_x\ne v_y breaks the C_{3z} symmetry and the trigonal warping term breaks both the P and T symmetry for each valley.

By diagonalizing the Hamiltonian Eq. (3), we obtain the the energy dispersion as

(4){\epsilon }_n(\mathit{k})=n\sqrt{d_x^2+d_y^2+d_z^2},

where we denote n=\pm for the conduction and valence bands, respectively, and d_x={\lambda }_1{k}_y^2-{\lambda }_2{k}_x^2+\xi v_x{k}_x, d_y=v_y{k}_y+2\xi {\lambda }_2{k}_x{k}_y, and d_z=\frac{\mathrm{\Delta }}{2}. The corresponding eigenstates are given by

(5){\mathrm{\Psi }}_{\pm }(\mathit{k})={\displaystyle \frac{1}{\sqrt{2\left|{\mathit{d}}_k\right|({\mathit{d}}_k\mp d_z)}}}\left(\begin{array}{c}{d}_x-\mathrm{i}{d}_y\\ \pm \left|{\mathit{d}}_k\right|-d_z\end{array}\right),

where {\mathit{d}}_k=(d_x,d_y,d_z).

In this work, we are interested in the second-order dc nonlinear Hall conductivity. When applying an ac electric field \mathit{E}(\omega )=\mathrm{\Re }(\mathit{E}{\mathrm{e}}^{\mathrm{i}\omega t}), the induced dc nonlinear current can be expressed as

(6)j_a=-e{\displaystyle \sum _{n,\mathit{k}}{\displaystyle \frac{{\mathrm{d}}^2\mathit{k}}{(2\pi {)}^2}}{f}_n(\mathit{k})v_a^n(\mathit{k}),}

where f_n(\mathit{k}) and v_a^n(\mathit{k})=(e/\hslash )\mathit{E}\times {\mathbf{\Omega }}_n are distribution function and anomalous Hall velocity, respectively. Here, The Berry curvature is given by

(7){\mathbf{\Omega }}_n=2e\underset{m\ne n}{Im{\displaystyle \sum }}{\displaystyle \frac{[A_{nm}^a{A}_{mn}^b]}{{\epsilon }_n-{\epsilon }_m(\mathit{k})}}.

Usually, there are two mechanisms contributing to the nonlinear Hall current. One is the electric-field correction to distribution function f_n(\mathit{k})\mathbf{⟶}e\tau E_b{\mathrm{\partial }}_{k_b}{f}_0({\epsilon }_n), which is given by the Boltzmann equation under the relaxation time approximation. Here, \tau denotes the scattering time and f_0({\epsilon }_n)=1/\left[1+e^{\left({\epsilon }_n-\mu \right)/k_{B}T}\right] is the equilibrium Fermi−Dirac distribution with \mu the chemical potential and T the temperature. When denoting j_a={\chi }_{a;bc}{E}_b{E}_c^{\ast } and performing the integration by parts, one can find the BCD nonlinear Hall conductivity as [3]

(8){\chi }_{a;bc}^{BCD}=-g_s{\displaystyle \frac{e^3\tau }{{\hslash }^2}}{ϵ}_{abd}{\displaystyle \sum _{n,\mathit{k}}({\mathrm{\partial }}_c{\mathbf{\Omega }}_n^d)f_0({\epsilon }_n),}

where {ϵ}_{abc} is the Levi−Civita tensor, g_s=2 accounts for spin degeneracy, and a,b,c=(x,y,z) are Cartesian indices, respectively describing the current direction (a) and electric field direction (b,c).

The other is electric-field correction to the Berry curvature in the anomalous Hall velocity from interband coherence, namely, v_a^n(\mathit{k})=(e/\hslash )\mathit{E}\times {\mathbf{\Omega }}_n^{\mathit{E}}, where the Berry curvature {\mathbf{\Omega }}_n^{\mathit{E}}={\mathbf{\nabla }}_{\mathit{k}}\times {\mathit{A}}_n^{\mathit{E}} is corrected by an electric-field-corrected Berry connection [5, 35, 40, 41] {\mathit{A}}_{n;a}^{\mathit{E}}=G_n^{ab}{E}_b. Here,

(9)G_n^{ab}=e{\displaystyle \sum }_{m\ne n}{\displaystyle \frac{Re[A_{nm}^a{A}_{mn}^b]}{{\epsilon }_n-{\epsilon }_m}},

is called the QMD or the Berry connection polarizability (BCP) tensor. Submitting Eq. (9) to Eq. (6) and keeping the electric field up to {\mathit{E}}^2 term, one can obtain the corresponding intrinsic nonlinear Hall conductivity, given by [40]

(10){\chi }_{a;bc}^{BCP}=-g_s{\displaystyle \frac{e^2}{2\hslash }}{\displaystyle \sum }_{n,\mathit{k}}(2{\mathrm{\partial }}_a{G}_n^{bc}-{\mathrm{\partial }}_b{G}_n^{ac}-{\mathrm{\partial }}_c{G}_n^{ab})f_0({\epsilon }_n).

In these nonlinear Hall effect, it is expected that the orbital magnetization will play a role. In the semiclassical picture, a Bloch electron is modeled by a wave packet in a Bloch band, which is found to rotate about its center of mass in general, yielding an orbital magnetic moment given by

(11){\mathit{m}}_n(\mathit{k})=-\mathrm{i}{\displaystyle \frac{e}{2\hslash }}⟨{\mathrm{\nabla }}_{\mathit{k}}{\mathrm{\Psi }}_n(\mathit{k})|\times \left[H(\mathit{k})-{\epsilon }_n(\mathit{k})\right]|{\mathrm{\nabla }}_{\mathit{k}}{\mathrm{\Psi }}_n(\mathit{k})⟩.

In the presence of a weak magnetic field B, the electron energy band is corrected to be {\epsilon }_n(\mathit{k})⟶{\epsilon }_n(\mathit{k})-{\mathit{m}}_n(\mathit{k})\cdot \mathit{B}. For an equilibrium ensemble of electrons, taking the differential of the total energy with respect to \mathit{B}, we can obtain the oribtal magnetization at zero magnetic field to be [42]

(12)\mathit{M}=2{\displaystyle \sum _{n,\mathit{k}}f({\epsilon }_n)[{\mathit{m}}_n(\mathit{k})+{\displaystyle \frac{e{\mathbf{\Omega }}_n(\mathit{k})}{\hslash }}(\mu -{\epsilon }_n(\mathit{k}))].}

For a two-dimensional system under consideration, the OMM is always perpendicular to the material plane and only the out-of-plane orbital magnetization M_z survives.

3 Results and discussion

3.1 Equilibrium OMM induced linear valley Hall effect

Before performing the calculation of nonlinear transports, we firstly discuss the relation between the linear Hall effect and the equilibrium orbit magnetization M_{eq}(\mathit{k}). The equilibrium OMM M_{eq}(\mathit{k}) can be obtained from Eq. (12) with f(\mathit{k})=f_0(\mathit{k}) chosen as the Fermi−Dirac distribution function, and so the OMM M_{eq}(\mathit{k}) is independent of electric field \mathit{E}. In Fig. 1(a) we present the distribution of the OMM M_{eq}(\mathit{k}) along k_x direction for conduction band (n=+) and K valley (\xi =+). It is evident that M_{eq}(\mathit{k}) always emerges, regardless of the existence of strain. Importantly, M_{eq}(\mathit{k}) is mainly concentrated around the valley regions and takes opposite sign in two valleys. We can integrate M_{eq}(\mathit{k}) over \mathit{k} around each valley to obtain the valley orbital magnetization M_{eq}(\xi ), and plot it as a function of chemical potential \mu in Fig. 1(b). Interestingly, M_{eq}(\xi ) has the same magnitude but with opposite sign at two valleys \mathit{K} and {\mathit{K}}^{\mathrm{\prime }}. Thus, the carriers from different valleys suffer from the opposite magnetization M_{eq}(\mathit{K})=-M_{eq}({\mathit{K}}^{\mathrm{\prime }}) and move towards opposite edge of the sample. It is just this valley-contrasting orbital magnetization that leads to the well-known valley Hall effect [25].

In Fig. 1(c), we plot the valley-resolved linear Hall conductivity {\chi }_{yx}^{\xi }, defined by j_y^{\xi }={\chi }_{yx}^{\xi }{E}_x, as a function of chemical potential \mu. It is evident that due to {\chi }_{yx}^{\xi } from each valley remaining finite and opposite sign, as indicated by the red and blue curves, the complete cancellation makes the charge Hall conductivity {\chi }_{yx}^c={\chi }_{yx}^K+{\chi }_{yx}^{K^{\mathrm{\prime }}} vanishes. Instead, this valley-contrasting response results in a finite valley Hall conductivity {\chi }_{yx}^v={\chi }_{yx}^K-{\chi }_{yx}^{K^{\mathrm{\prime }}}. This is a result of time-reversal invariant. In Fig. 1(d), we depict the dependence of the valley Hall conductivity {\chi }_{yx}^v on the staggered potential \mathrm{\Delta }. Obviously, {\chi }_{yx}^v has wider zero-conductivity plateau with the increase of energy gap induced by \mathrm{\Delta } and vanishes in the presence of inversion symmetry, i.e., \mathrm{\Delta }=0.

3.2 Nonequilibrium shift OMM induced nonlinear charge Hall effect

Above, we discuss the relation of the valley orbital magnetization with linear valley Hall effect. It is interesting to extend this scenario to the nonlinear situation. When considering the OMM corrected up to the first order in the electric field, we have two choices: the correction of distribution function or the correction of \mathit{m}(\mathit{k}) and \mathbf{\Omega }(\mathit{k}). Firstly, we discuss the first case. For the orbital magnetization in Eq. (12), according to Boltzmann equation we can correct the distribution function f(\mathit{k}) up to the first-order in the electric field, which is approximated as f(\mathit{k})\approx f_0[\epsilon (\mathit{k})]+(e/\hslash )\tau \mathit{E}\cdot {\mathrm{\partial }}_{\mathit{k}}{f}_0[\epsilon (\mathit{k})]. Thus, from Eq. (12) the electric-field corrected magnetization can be expressed as

(13){\mathit{M}}_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}}={\displaystyle \frac{2e\tau }{\hslash }}{\displaystyle \int {\displaystyle \frac{{\mathrm{d}}^2\mathit{k}}{(2\pi {)}^{{}_2}}}\mathit{E}\cdot {\mathrm{\partial }}_{\mathit{k}}{f}_0[\epsilon (\mathit{k})]\mathit{m}(\mathit{k}).}

At low temperatures, we use {\mathrm{\partial }}_{\mathit{k}}{f}_0[\epsilon (\mathit{k})]=\frac{\mathrm{\partial }\epsilon (\mathit{k})}{\mathrm{\partial }\mathit{k}}\delta (\epsilon (\mathit{k})-\mu ) to take a calculation. This correction originates from the shift of Fermi surface by the applied in-plane electric field, attributing to intraband contribution. Also, \mathit{E}-corrected {\mathit{M}}_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}} has only the out-of-plane component.

In Fig. 2, we depict the distribution of the orbit magnet momentum M_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}}(\mathit{k}) in momentum space (k_x,k_y). In the presence of trigonal warping effect but without strain [seeing Fig. 2(a)], both the Fermi surface and M_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}}(\mathit{k}) for the \mathit{K} valley become trigonal-like, namely C_{3z} rotation symmetry. After integration of \mathit{k}, the orbital magnetization M_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}}=0, which perfectly cancels all contributions. Meantime, in the absence of trigonal warping effect (i.e., {\lambda }_0=0), although the shapes of the energy band and M_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}}(\mathit{k}) for the \mathit{K} valley are distorted by strain, they are the odd function of \mathit{k}. Only when both the strain and warping effect coexist, the fermi surface and M_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}}(\mathit{k}) are distorted to be asymmetric, as shown in Fig. 2(b) where a uniaxial strain {\epsilon }_{xx} is applied along x-axis, namely, along the zigzag direction of the crystal. Thus, after integrating of of \mathit{k}, finite M_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}} will appear. As \mathit{E}-corrected OMM M_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}}(\mathit{k}) is also mainly concentrated around two valleys, as shown in Fig. 2(c), we still can integrate M_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}}(\mathit{k}) around each valley to obtain the the \mathit{E}-corrected valley orbital magnetization M_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}}(\xi ), which is plotted in Fig. 2(d) as a function of chemical potential \mu. Notice that M_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}}(\mathit{k}) takes the same sign at two valleys, distinguishing from the equilibrium OMM M_{eq}(\mathit{k}) which has opposite sign. As a result, the valley orbital magnetization M_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}}(\mathit{K})=M_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}}({\mathit{K}}^{\mathrm{\prime }}) is strictly guaranteed. Thus, the carriers from different valleys suffer from the same orbital magnetization and move towards the same edge of the sample. We can understand the valley dependence from the viewpoint of symmetry. When taking an operation of spatial inversion (P), two valleys exchange each other \mathit{K}\leftrightarrow {\mathit{K}}^{\mathrm{\prime }}, and \mathit{k}\to -\mathit{k} where \mathit{k} is measured from the valley. It is easy to check that both \mathit{m}(\mathit{k}) and {\mathrm{\partial }}_{\mathit{k}}{f}_0[\epsilon (\mathit{k})] is P-odd. Thus, their product is P-even, and after integrating over \mathit{k}, the distribution-function-induced shift magnetization in Eq. (13) is valley-independent, producing a nonlinear charge Hall response. Figures 2(c) and (d) also show that the valley orbital magnetization increases with the enhancement of strain.

We calculate the BCD nonlinear Hall conductivity with Eq. (8), and in Fig. 3(a) the valley-resolved nonlinear Hall conductivity {\chi }_{y;xx}^{\xi } is plotted as a function of chemical potential \mu. It is evident that {\chi }_{y;xx}^{\xi } from each valley has the same size and the same sign. Thus, there is no nonlinear valley Hall current {\chi }_{a;bc}^v={\chi }_{a;bc}^K-{\chi }_{a;bc}^{K^{\mathrm{\prime }}}, but exists a nonlinear charge Hall current, defined by {\chi }_{a;bc}^c={\chi }_{a;bc}^K+{\chi }_{a;bc}^{K^{\mathrm{\prime }}}. Physically, due to the valley-independent M_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}}(\mathit{k}), the carriers from different valleys suffer from the same magnetization and move towards the same edge of the sample. In contrast to the valley-contrasting orbital magnetization in linear valley Hall effect shown in Fig. 1, here the valley-independent orbital magnetization interprets the formation of BCD nonlinear Hall effect. We are known that the BCD-induced nonlinear Hall effect has been extensively received attention, but the underlying physics remains to be uncovered. To be specific, here the nonlinear Hall conductivity {\chi }_{y;xx}^{\xi } exhibits a second-order response to the electric field, and we can attribute it to a two-step process: One electric field E_x induces an orbital magnetization M_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}}(\xi ) and then the other electric field E_x generates the anomalous Hall effect in the presence of the orbital magnetization.

In addition, in Fig. 3, one can notice that {\chi }_{y;xx}^{\xi } exhibits a non-monotonic behavior, antisymmetric with respect to \mu, characterized by two peaks in either side of the Dirac point, followed by a suppression for \mu deviating from the Dirac point, which is essentially determined by the BCD. When chemical potential \mu gets close to the Dirac point, the fermi surface manifests itself as a circle and the trigonal warping is not apparent, resulting in an almost vanishing nonlinear anomalous effect. As increasing the value of |\mu |, on one hand, the Berry curvature will decrease and tend to weaken the signal of nonlinear Hall effect, and on the other hand, the fermi surface will start to deviate from circle and the trigonal warping effect gradually becomes more profound which tends to enhance the nonlinear Hall effect. As a consequence, the competition between this two mechanisms cause the non-monotonic behavior.

Figure 3(b) shows that nonlinear charge Hall conductivity has direction selectivity. As one applies an external electric field E_y along the armchair direction (i.e., y-direction), {\chi }_{x;yy}^{\xi } and {\chi }_{x;yy}^c vanish. It is obvious that the threefold rotation C_{3z} about k_z is broken by the strain but the mirror symmetry M_{ky} is still survived. As we know, the BCD vector, defined by D_a=\int \frac{\mathrm{d}\mathit{k}}{(2\pi {)}^{{}_2}}{f}_0(\mathit{k})\frac{\mathrm{\partial }{\mathrm{\Omega }}_z(\mathit{k})}{\mathrm{\partial }{k}_a}, always aligns with the mirror-symmetry axis, i.e., along the zigzag (x direction). Thus, {\chi }_{y;xx}^{\xi } is finite for the electric field along the zigzag direction whereas {\chi }_{x;yy}^{\xi } vanishes for the electric field along the armchair direction (y direction). In Fig. 3(c), we show that the nonlinear charge Hall conductivity {\chi }_{y;xx}^c increases with the enhancement of strain |{\epsilon }_{xx}| and vanishes for {\epsilon }_{xx}=0. The underlying physics lies that as the strain increases, the electronic dispersion becomes more anisotropic, which leads to a pronounced enhancement of the orbital magnetization. Interestingly, when changing the strain from tensile strain ({\epsilon }_{xx}>0) to compressive strain (), the nonlinear charge Hall conductivity {\chi }_{y;xx}^c inverses the sign, which is attributed to the inversion of the orbital magnetization M_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}}. The nonlinear charge Hall conductivity {\chi }_{y;xx}^c also becomes significant with the enhancement of warping effect {\lambda }_0, as shown in Fig. 3(d). In addition, for the gapless energy with \mathrm{\Delta }=0, the orbital magnetization strictly remains zero and so nonlinear charge Hall current is zero.

3.3 Nonequilibrium intrinsic OMM induced nonlinear valley Hall effect

Finally, we discuss anther case (2): The orbital magnetization in Eq. (12) corrected up to the first order in the electric field through \mathit{m}(\mathit{k}) and \mathbf{\Omega }(\mathit{k}) has been derived in the semiclassical theory framework [24, 43, 44]. Due to the equilibrium distribution f(\mathit{k})=f_0(\mathit{k}) remaining the Fermi-Dirac distribution function, the orbital magnetization derived from Eq. (12) is intrinsic, independent on impurity scattering. Under general two-band model, this intrinsic magnetization reduces to be

(14)M_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{r}}={\displaystyle \frac{e}{2}}{\displaystyle \sum }_{n,m\ne n;\mathit{k}}{f}_0(\mathit{k})\left[{\displaystyle \frac{\mathrm{\partial }{\epsilon }_n(\mathit{k})}{\mathrm{\partial }{k}_b}}{G}_n^{zd}+{\displaystyle \frac{e}{\hslash }}{ϵ}_{zbc}{\displaystyle \frac{\mathrm{\partial }Re[A_{nm}^c{A}_{mn}^d]}{\mathrm{\partial }{k}_b}}\right]E_d,

where the band-resolved quantum metric G_n^{zd} is defined in Eq. (9).

In Fig. 4(a), we depict the \mathit{E}-corrected valley orbital magnetization M_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{r}}(\xi ) as a function of chemical potential \mu. Interestingly, this electric field-induced M_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{r}}(\xi ) for each valley is also equal in magnitude but has the opposite sign, similar to the case of linear valley Hall effect in Fig. 1. Consequently, this valley-contrasting M_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{r}}(\mathit{K})=-M_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{r}}({\mathit{K}}^{\mathrm{\prime }}) forces the carriers from different valleys moving towards the opposite edge of the sample, causing the nonlinear valley Hall effect without the nonlinear charge Hall effect, as shown in Fig. 4(b). Here, the QMD nonlinear Hall conductivity is calculated with formula (10). Since the QMD in Eq. (14) is P-odd while the Fermi distribution f_0[\epsilon (\mathit{k})] is P-oven, their product is P-odd, leading to the valley-contrasting intrinsic magnetization in Eq. (14). Notice that the nonlinear valley Hall effect was reported in tilted massless Dirac fermions [24]. Despite of the different Hamiltonian, the underlying physics is the same for tilt and warping, both of which break the T and P symmetry for each valley but remain T and P symmetry globally. Once this symmetry is satisfied, the nonlinear valley Hall effect will appears. In these systems, the valley-contrasting orbital magnetization is essentially contributed by the BCP. Unlike the BCD nonlinear charge Hall effect, this second-order valley Hall effect does not always require gapped dispersion, as shown in Fig. 4(d) where \mathrm{\Delta } only changes the magnitude but is not decisive. As the nonlinear valley Hall effect {\chi }_{y;xx}^v exhibits a non-monotonic behavior, it is symmetric with respect to \mu.

In Fig. 4(c), it is noted that only {\chi }_{x;yy}^v components of nonlinear valley Hall conductivity is finite while {\chi }_{y;xx}^v vanishes, in contrast to the BCD induced nonlinear charge Hall effect. For the Hamiltonian Eq. (1), both nonlinear charge and valley Hall conductivities coexist but along orthogonal direction, which can be departed in experimental measurements. Due to the strain enhancing the breaking of symmetry, the nonlinear valley Hall conductivity {\chi }_{x;yy}^v also becomes significant with the enhancement of strain, and changes the sign when the train changes from {\epsilon }_{xx}>0 to .

4 Conclusion

In summary, we have presented a unified theoretical framework to understand the nonlinear valley Hall effect and nonlinear charge Hall effect in strained monolayer graphene. As we know, both of them can appear in the case of time-reversal symmetry, violating the theory of conventional linear Hall effect. Usually, the valley Hall effect is interpreted with OMM whereas nonlinear charge Hall effect is understood with BCD. In this work, we provide deeper insights into the nonlinear transports and find that both nonlinear valley and nonlinear charge Hall effects can be interpreted with the OMM. We employ an effective low-energy Hamiltonian of strained graphene, which is derived from a tight-binding model with strain-modified hopping parameters. Since strain induced anisotropic velocities and band-warping terms break the inversion and rotation symmetry, the nonlinear valley and charge Hall effect emerges. We demonstrate that the equilibrium OMM is valley-contrasting which contributes to the linear valley Hall current, the intrinsic OMM, originating from Berry curvature linearly corrected by electric field, is also valley-contrasting which contributes to the nonlinear valley Hall current, and the shift OMM, originating from Fermi distribution function linearly corrected by electric field, is valley-independent which contributes to the nonlinear BCD Hall current. Therefore, the nonlinear BCD Hall current and nonlinear valley current essentially have the same physics and the dependence of valley index of the OMM determines which nonlinear Hall effect emerges. Meantime, the observed valley magnetization increases with the increasing magnitude of strain, reversibly turns on and off, and flips the sign, depending on the direction of strain, which provides a route to strain-engineered control of OMM and so the nonlinear transports. Notice that in our investigation, we employ the strained graphene only as an example to shed light on the unified framework for understanding both the nonlinear charge Hall and nonlinear valley Hall effects under the two-step mechanism. Naturally, this results are general and can be applied to the other noncentrosymmetric or ferrovalley two-dimensional materials where two-type nonlinear Hall effects coexist.

Experimentally, the orbital magnetization can be directly imaged via Magneto-optical Kerr effect (MOKE) microscopy as in Ref. [45]. The MOKE describes the rotation of polarization of linearly polarized light when reflected by a material in the presence of a magnetic field or magnetization, especially for out-of-plane magnetization in 2D materials. Here, both of the shift-induced and intrinsic orbital magnetization are out-of-plane and the MOKE microscopy is accessible. In addition, The shift-induced magnetization can be probed indirectly through standard second-harmonic transport experiments [4, 46] or orbital transfer torque exerted on the adjacent magnetizations as reported in recent works [47–50]. In contrast, the nonlinear valley Hall effect driven by intrinsic orbital magnetization cannot be directly measured due to no net charge current. But, one can excite electrons into a specific valley by illuminating with circularly polarized light, causing a finite anomalous Hall voltage whose sign is controlled by the helicity of the light as in Ref. [51].

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