Magnus Hall Effect

TL;DR

This paper predicts a novel linear response Hall effect in time-reversal-invariant systems caused by a quantum Magnus effect, linking the Hall response to Berry curvature distribution without magnetic fields.

Contribution

It introduces the concept of the Magnus Hall effect, a new type of Hall response arising from self-rotating electron wavepackets in electric fields, expanding understanding of Hall phenomena.

Findings

Magnus Hall conductance relates to Berry curvature on the Fermi surface

The effect occurs at zero magnetic field in time-reversal-invariant systems

The phenomenon is described in the ballistic transport limit

Abstract

A new type of linear response Hall effect is predicted in time-reversal-invariant systems with built-in electric field at zero magnetic field. The Hall response results from a quantum Magnus effect where a self-rotating Bloch electron wavepacket moving under electric field develops an anomalous velocity in the transverse direction. We show that in the ballistic limit the Magnus Hall conductance measures the distribution of Berry curvature on the Fermi surface.

Equations

$$\dot{\mathbf{r}}=\frac{1}{\hslash }\frac{\partial {ϵ}_{\mathbf{k}}}{\partial \mathbf{k}}-\frac{1}{\hslash }\mathbf{\Omega }\times \frac{\partial U}{\partial \mathbf{r}},\dot{\mathbf{k}}=-\frac{1}{\hslash }\frac{\partial U}{\partial \mathbf{r}}$$

$$\Delta y_A$$

$$=\frac{1}{\hslash v_x}\Omega ({\mathbf{k}}_0)\Delta U$$

$$\frac{\partial f}{\partial t}+\dot{\mathbf{r}}\cdot \frac{\partial f}{\partial \mathbf{r}}+\dot{\mathbf{k}}\cdot \frac{\partial f}{\partial \mathbf{k}}=0$$

$$ϵ(\mathbf{k},\mathbf{r})={ϵ}_{\mathbf{k}}+U(x)$$

$$f_0(\mathbf{k},\mathbf{r})={\left(e^{\beta (ϵ(\mathbf{k},\mathbf{r})-\bar{\mu })}+1\right)}^{-1}$$

$$\frac{1}{\hslash }\frac{\partial f_1}{\partial x}\frac{\partial {ϵ}_k}{\partial k_x}-\frac{1}{\hslash }\frac{\partial U}{\partial x}\frac{\partial f_1}{\partial k_x}=0$$

$$f_1(\mathbf{k},\mathbf{r})=\{\begin{array}{ll}\left(-\frac{\partial f_0}{\partial {ϵ}_k}\right)\Delta \bar{\mu }& v_x(\mathbf{k})>0\\ 0& v_x(\mathbf{k})<0\end{array}$$

$$j_x$$

$$j_y$$

$$j_y^0$$

$$j_H$$

$$G_H=\frac{e^2}{h}\frac{\Delta U}{2\pi }\underset{v_x(\mathbf{k})>0}{\int }{d}^{2}k\Omega (\mathbf{k})\delta \left({ϵ}_{\mathbf{k}}-\mu \right)$$

$$H(\mathbf{k})=A{k}^2+\left(B{k}^2+\delta \right){\sigma }_z+v_y{k}_y{\sigma }_y+D{\sigma }_x.$$

$$H_s=\left(\begin{array}{cc}\Delta & sv{k}_{-s}-\lambda k_s^2\\ sv{k}_s-\lambda k_{-s}^2& -\Delta \end{array}\right)$$

$$\frac{\partial f}{\partial t}+\dot{\mathbf{r}}\cdot \frac{\partial f}{\partial \mathbf{r}}+\dot{\mathbf{k}}\cdot \frac{\partial f}{\partial \mathbf{k}}=-\frac{f-f_0}{\tau }$$

$$\frac{1}{\hslash }\frac{\partial f}{\partial s}\frac{\partial {ϵ}_k}{\partial \mathbf{k}}\cdot \hat{n}+\frac{1}{\hslash }F(s)\frac{\partial f}{\partial \mathbf{k}}\cdot \hat{n}=-\frac{f-f_0}{\tau }$$

$$\frac{1}{\hslash }\frac{\partial f_1}{\partial s}\frac{\partial {ϵ}_k}{\partial \mathbf{k}}\cdot \hat{n}+\frac{1}{\hslash }F(s)\frac{\partial f_1}{\partial \mathbf{k}}\cdot \hat{n}=-\frac{f_1}{\tau }$$

$$\frac{\partial f_1}{\partial s}=-\frac{f_1}{v_n\tau }$$

$$f_1(\mathbf{k},\mathbf{r})=A(\mathbf{k})e^{-\frac{s}{v_n\tau }}+B(\mathbf{k})$$

$$f_1(\mathbf{k},\mathbf{r})=\{\begin{array}{ll}\left(-\frac{\partial f_0}{\partial {ϵ}_k}\right)\Delta \bar{\mu }{e}^{-\frac{s}{v_n\tau }}& v_n(\mathbf{k})>0\\ 0& v_n(\mathbf{k})<0\end{array}$$

$$H(\mathbf{k})=A{k}^2+\left(B{k}^2+\delta \right){\sigma }_z+v_y{k}_y{\sigma }_y+D{\sigma }_x$$

$$U(x)=-\Delta U\tanh \left(\frac{x/L-x_c}{s}\right)$$

$$I_x=G_{xx}\Delta V_{12}^{(1)}+G_{xy}\Delta V_{34}^{(1)}$$

$$0=G_{yx}\Delta V_{12}^{(1)}+G_{yy}\Delta V_{34}^{(1)}$$

$$0=G_{xx}\Delta V_{12}^{(2)}+G_{xy}\Delta V_{34}^{(2)}$$

$$I_y=G_{yx}\Delta V_{12}^{(2)}+G_{yy}\Delta V_{34}^{(2)}$$

Full text

Introduction.— Studies of various Hall effects led to significant progress throughout the history of solid state physics. Starting with the classical Hall effectHall (1879), anomalous Hall effect Nagaosa et al. (2010), spin Hall effect Sinova et al. (2015), thermal Hall effect Banerjee et al. (2018); Kasahara et al. (2018), quantum Hall effect v. Klitzing et al. (1980), quantum spin Hall effect Kane and Mele (2005) and quantum anomalous Hall effect Haldane (1988); Chang et al. (2013); Liu et al. (2016) have been discovered. Among these, classical and anomalous Hall effects appear in time-reversal-breaking systems, where an applied electric field induces a transverse charge current.

An intrinsic contribution to anomalous Hall effect is associated with Berry curvature, a fundamental ingredient of modern band theory derived from electron’s wavefunction within the unit cell Karplus and Luttinger (1954). When a Bloch electron is accelerated under an electric field, the redistribution of electron density within the unit cell changes with increasing momentum, thus giving rise to an anomalous velocity proportional to Berry curvature. In time reversal breaking systems, total Berry curvature of occupied states can be nonzero, resulting in intrinsic anomalous Hall effect. The impact of Berry curvature on transport phenomena has attracted tremendous interest Xiao et al. (2010); Moore and Orenstein (2010); Morimoto et al. (2016); Morimoto and Nagaosa (2016); Chaudhary et al. (2018); Rudner and Song (2018).

On the other hand, there exists a large class of time-reversal-invariant, inversion-breaking materials which feature large Berry curvature $\bm{\Omega}(\bm{k})$ in momentum space, especially near the gap edge or band crossing points. Examples include two-dimensional transition metal dichalcogenides (TMDs) Qian et al. (2014), graphene multilayers McCann and Koshino (2013); Rozhkov et al. (2016) and heterostructures Yankowitz et al. (2012), topological insulator surface states Fu (2009) and Weyl semimetals Hasan et al. (2017); Armitage et al. (2018); Yan and Felser (2017). Due to time reversal symmetry, the distribution of Berry curvature satisfies $\bm{\Omega}(\bm{k})=-\bm{\Omega}(-\bm{k})$. It is an intriguing question whether such a distribution of Berry curvature with zero total can lead to any interesting phenomena in charge transport.

In this work, we demonstrate a new type of linear-response Hall effect induced by Berry curvature and built-in electric field in mesoscopic systems under time-reversal-symmetric condition. We consider electron transport in a Hall bar device made of a 2D material, such as bilayer graphene or transition metal dichalcogenide. In our setup, source and drain regions have different carrier densities, which can be achieved by local bottom gates. The difference in electrostatic potential energy due to bottom gates $U_{s}-U_{d}\equiv\Delta U$ is accompanied by a built-in electric field in the junction at equilibrium, as shown in Fig. 1(a). We study electrical current in linear response to applied bias voltage $V_{sd}$. Since electrons moving from source to drain have nonzero net velocity, their wave packets can carry orbital angular momentum and nonzero net Berry curvature. The motion of chiral Bloch electrons under built-in electric field leads to a quantum analog of Magnus effect: as an electron traverses the junction, its center of wavepacket acquires a transverse shift. This in turn gives rise to a transverse current linearly proportional to the bias voltage, i.e., a Hall effect. We term this phenomenon “Magnus Hall effect”. It occurs in nonmagnetic systems at zero magnetic field, but relies on the built-in electric field. We show that in the ballistic limit, the Hall conductance is directly related to the Berry curvature distribution of the underlying material.

Magnus Hall effect is intimately related to nonlinear Hall effect, where recent theory predicted that in time-reversal-invariant materials the dipole moment of Berry curvature, i.e., Berry curvature dipole, can induce a nonlinear Hall effect, where the transverse current depends quadratically on the applied electric field Sodemann and Fu (2015). This phenomenon has been subsequently observed in bilayer WTe2 Ma et al. (2019); Kang et al. (2019). In the case of nonlinear Hall effect, reversing the electric field doesn’t change the transverse current. In Magnus Hall effect, transverse current is preserved under reversing both source-drain voltage and the direction of built-in electric field as presented in Fig.1(b). In an alternative scenario where the electric field arises from the source-drain voltage $V_{sd}$ itself instead of bottom gates, the transverse current becomes second order in $V_{sd}$, resulting in nonlinear Hall effect. Our work thus opens a pathway to “Hall diodes” for high-frequency nonlinear transport based on quantum materials that have a significant distribution of Berry curvature.

Electronic Magnus effect.— First we shall consider an electron wave-packet which travels inside a sample. The sample has a short segment (between $x=0$ and $x=L$) within which there is spatially varying potential energy $U(x)$ arising from different gate voltages on opposite sides (this corresponds to built-in electric field $\mathbf{E}=\frac{1}{e}\frac{\partial U}{\partial x}\hat{x}=E_{x}(x)\hat{x}$). Inside this region, motion of the wave-packet is described by the semiclassical equations of motion Xiao et al. (2010):

[TABLE]

Since the built-in electric field is small, wave packet momentum $\mathbf{k}$ does not change substantially under acceleration. Therefore, the transit time through the electric field region of an incident electron with velocity $(v_{x},v_{y})$ is simply $t=L/v_{x}$. During this time, the electron with $v_{y}\neq 0$ will also travel in the $y$ direction. Importantly, between $x=0$ and $x=L$ there will be an additional displacement of wave packet along $y$ due to anomalous velocity. This displacement is given by:

[TABLE]

with $\Delta U=-\int_{0}^{L}\frac{\partial U}{\partial x}dx$ being the difference in potential energy. Therefore, an electron wave packet with non-zero Berry curvature moving through the region of electric field will acquire an additional shift in the direction perpendicular to the electric field as schematically shown in Fig. 1(a). This in turn leads to a current density in $y$ direction by integrating single wave-packet contribution $-e\Delta y_{A}/t=-e/(\hbar L)\Omega(\mathbf{k}_{0})\Delta U$ over occupied states. This transverse current vanishes in equilibrium. However, in the current carrying state, the modes with positive and negative velocity are not equally occupied. Since in systems with time-reversal symmetry Berry curvature is opposite for $\mathbf{k}$ and $-\mathbf{k}$ states (which also have opposite velocities), this effect can lead to a Hall current even in nonmagnetic materials. To see more clearly how the Hall current arises here, we employ Boltzmann transport equation.

Boltzmann transport equation solution.— To describe mesoscopic electron transport in a 2D system we use a collisionless Boltzmann equation (BE):

[TABLE]

where $f(\mathbf{k},\mathbf{r})$ is the occupation distribution function. Later we shall extend the discussion to non-ballistic case with finite relaxation time. We are looking for a stationary state distributions, so $\frac{\partial f}{\partial t}=0$. We solve the Boltzmann equation in a geometry of a stripe of infinite width in $y$ direction and finite length $L$ in $x$ direction. At $x<0$ and $x>L$ we have a source and a drain. Since the system is translationally invariant in $y$ direction, the stationary distribution function will be independent of $y$.

The wave packets evolve according to semiclassical equations given by Eq.(1). The energy of the electrons in the segment is:

[TABLE]

where $\epsilon_{\mathbf{k}}$ is the band energy. In equilibrium, the solution is given by Fermi-Dirac distribution with constant electrochemical potential $\bar{\mu}$, but with spatially changing energy (4) and can be expressed as:

[TABLE]

$f_{0}(\mathbf{k},\mathbf{r})$ is a solution of Boltzmann equation (3) as can be verified using Eqs.(1). This solution guarantees that no current is flowing in the system as at each energy and position the number of states with $\mathbf{k}$ and $-\mathbf{k}$ is equal.

To obtain a solution that corresponds to a steady current flow, we apply a small bias $V_{sd}$ in a form of imbalance of electrochemical potentials between source and drain, so that $\bar{\mu}_{D}=\bar{\mu}$ and $\bar{\mu}_{S}=\bar{\mu}-eV_{sd}=\bar{\mu}+\Delta\bar{\mu}$. We have to solve BE with boundary conditions that take into account the presence of reservoirs at $x=0$ and $x=L$. Taking into consideration the device geometry of our system, we now look for a solution in lowest order of the perturbation in the electrochemical potential imbalance. To achieve this, we write $f=f_{0}+f_{1}$, where $f_{1}$ is the nonequilibrium part first order in the perturbation $\Delta\bar{\mu}$. This gives an equation for $f_{1}$:

[TABLE]

Since we assumed that $U(x)$ is slowly varying, the second term on the left hand side is small, so we drop it and we arrive at $\frac{\partial f_{1}}{\partial x}=0$. Therefore, $f_{1}(\mathbf{k},\mathbf{r})$ only depends on $\mathbf{k}$. We can now determine its form from the boundary condition. The larger electrochemical potential of the source region results in a surplus of electrons entering the system at $x=0$ interface with positive $v_{x}$ velocity and propagating across the device without any scattering in the ballistic limit. This boundary condition gives us:

[TABLE]

where $v_{x}(\mathbf{k})=\frac{1}{\hbar}\frac{\partial\epsilon_{k}}{\partial k_{x}}$. Equipped with this solution, we are able to calculate the longitudinal and Hall response of our system. We have then:

[TABLE]

[TABLE]

Here $j_{y}^{0}$ term arises from Fermi surface anisotropy and depends on its orientation relative to the direction of applied bias $\hat{x}$. Note that $j_{y}^{0}$ is independent of $\Delta U$. However, as we shall show later, this term can vanish due to symmetry.

To obtain the Hall current due to Berry curvature, we integrate the anomalous velocity contribution over the whole length of the device $I_{y}=\int_{0}^{L}dxj_{H}$ and then we can define Hall conductance as $G_{H}=-eI_{y}/\Delta\bar{\mu}$ and obtain at $T=0$:

[TABLE]

where the electrostatic potential energy difference across the junction $\Delta U=U_{S}-U_{D}$ is assumed to be small.

This equation is the main result of this work. First of all, in the limit of small $\Delta U$ the Hall response does not depend on the detailed spatial dependence of the potential energy. Second, the dependence on $\Delta U$ is linear and the Hall current is independent of the system size in the ballistic limit. The effect is reduced by finite relaxation time as we show in Supplementary Materials. Crucially, Magnus Hall current is determined by directional Berry curvature of electrons with positive velocity along the direction of the applied bias. Therefore, by varying the bias with respect to crystal axis, Magnus Hall conductance can be used to map Berry curvature distribution on the Fermi surface.

Model.— To demonstrate this effect explicitly, we turn to a concrete model, which breaks inversion symmetry, but preserves time-reversal symmetry. We have chosen a simple two band model with the Hamiltonian:

[TABLE]

It contains two massive Dirac cones which are tilted when $A\neq 0$. This model captures essential features of the tilted Dirac cones of topological crystalline insulator surface states Sodemann and Fu (2015) and low-energy band structure of 2D WTe2 Qian et al. (2014). More details of the model are presented in the Supplemental Materials.

We can now compute Berry curvature $\mathbf{\Omega}(\mathbf{k})$ distribution in the Brillouin zone for parameters $A=0,B=1,\delta=-0.25,v_{y}=1.0,D=0.1$, which is shown in the inset of Fig.2 for the valence band. While the total Berry curvature integrated over Brillouin zone is equal to 0, the distribution consists of two peaks of opposite signs, located at the Dirac points.

Furthermore, this model possesses mirror symmetry $M_{x}:H(k_{x},k_{y})\rightarrow H(-k_{x},k_{y})$, which guarantees that $\epsilon(k_{x},k_{y})=\epsilon(-k_{x},k_{y})$. For $D=0$, this model has additional mirror symmetry $M_{y}:H(k_{x},k_{y})=\sigma_{z}H(k_{x},-k_{y})\sigma_{z}$. This symmetry is broken for $D\neq 0$. However, because the model is also time-reversal-invariant, we have $\epsilon(-k_{x},k_{y})=\epsilon(k_{x},-k_{y})$, which causes the current density $j_{y}^{0}$ of Eq.(9) to vanish. Therefore, the Hall current will be determined solely by Magnus Hall contribution $j_{H}$ of Eq.(10) and $G_{H}$ can be calculated according to Eq. 11. Result of such a calculation is presented as the dashed curve in Fig.2 as a function of chemical potential. Even though the total integral of $\Omega(\mathbf{k})$ vanishes, because our result for $G_{H}$ only relies on the Berry curvature of Bloch states with $v_{x}>0$, it is non-zero. For the set of parameters used in the calculation, both bands are symmetric with respect to $E=0$ line and so the result for the conduction band is mirror image of the curve for valence band. We note that the direction of the Hall current is the same for both bands, because while the Berry curvature switches sign to opposite between two bands, the velocities for given $\mathbf{k}$ also change to opposite and as a results the integration occurs over regions with the same values of $\Omega(\mathbf{k})$.

These approximate results derived from the semiclassical Boltzmann transport approach can be compared with a numerical tight-binding simulation using Landauer-Buttiker method. Details of the calculation are in Supplemental Materials. All the numerical simulations have been performed using Kwant package Groth et al. (2014). We consider the chemical potential dependence of $G^{LB}_{H}$, which we present as the solid curve in Fig. 2 for $L=200$, $W=2400$ lattice sites and $\Delta U=0.05$. The curve is obtained by temperature broadening with $T=0.01$. The qualitative behavior of the semiclassical result is reproduced with an asymmetric peak positioned away from the band bottom. The difference can be attributed to the differences in geometry of the setups for numerical simulation (finite width leads injecting current into the system) and semiclassical calculation (infinite width of the setup).

Candidate materials.— To observe the Magnus Hall effect under time-reversal symmetry, several conditions must be satisfied. First of all, the underlying material must break the inversion symmetry in order to have non-zero Berry curvature in the Brillouin zone. Furthermore, Berry curvature of right and left moving modes must be asymmetric. For example, under time-reversal symmetry massive Dirac fermions have to appear in pairs with opposite sign of Berry curvature. If they are isotropic, there will be an equal number of right movers with both signs of $\Omega(\mathbf{k})$ in each Dirac valley and their contributions will cancel each other. However, in general the Dirac cones are not perfectly isotropic and perfect cancellation will not occur. Two examples of materials that satisfy this condition are monolayer graphene on hBN (sublattice symmetry is broken due to formation of Moire superlatice) Park et al. (2008); Yankowitz et al. (2012); Hunt et al. (2013); Yankowitz et al. (2018) and bilayer graphene with perpendicular electric field applied McCann and Fal’ko (2006); McCann and Koshino (2013); Rozhkov et al. (2016). In both cases trigonal warping introduces asymmetry between right and left movers in each valley. Moreover, these platforms support devices of high quality, which enable ballistic motion of electrons Du et al. (2008); Mayorov et al. (2011); Banszerus et al. (2016), beneficial for observing the predicted effect.

As an example we use a model of bilayer graphene with trigonal warping and perpendicular electric field that opens up a gap. Calculations are performed using low-energy Hamiltonian that describes both $K$ and $K^{\prime}$ valleys of bilayer graphene (labeled by $s=\pm 1$) McCann and Fal’ko (2006):

[TABLE]

where $k_{\pm}=k_{x}\pm ik_{y}$. Berry curvature distribution near both valleys is presented in the inset of Fig.3. The parameters used in the calculations are $\Delta=50$ meV, $v=10^{5}$ m/s and $\lambda=1/(2m^{*})$ with effective mass $m^{*}=0.033m_{e}$, $m_{e}$ being electron mass Mucha-Kruczyński et al. (2010); McCann and Koshino (2013). The Fermi surface at $\mu=-60$ meV is indcated by the green line, solid for $v_{x}=\frac{1}{\hbar}\frac{\partial\epsilon_{k}}{\partial k_{x}}>0$ and dashed for $v_{x}<0$. We can now use this to compute the Hall conductance as a function of chemical potential in the vicinity of the band edge, which is shown in Fig. 3 for $\Delta U=10$ meV. Our result demonstrates explicitly that Magnus Hall effect does not rely on Berry curvature dipole Sodemann and Fu (2015); Facio et al. (2018); Tsirkin et al. (2018); You et al. (2018); Zhang et al. (2018a, b); Shi and Song (2019) or the presence of skew scattering Isobe et al. (2018); Nandy and Sodemann (2019); König et al. (2018); Du et al. (2018) that are necessary conditions for nonlinear Hall effect.

Summary.— In this paper we have demonstrated existence of Magnus Hall effect in inversion symmetry breaking, but time reversal invariant systems that have non-zero Berry curvature. The effect relies on built-in electric field in the device and should be most pronounced in ballistic systems. The significance of the Magnus Hall effect is twofold. It opens a pathway to a new generation of current rectification devices. It also provides a much needed tool to map Berry curvature distribution of quantum materials in momentum space.

Acknowledgments.— We thank Hiroki Isobe and Su-Yang Xu for helpful discussions. This work was supported by DOE Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-SC0018945. MP was supported by Shell through the MIT Energy Initiative. LF was supported in part by a Simons Investigator Award from the Simons Foundation.