Planar Hall Effect in 2D Materials

• Planar Hall effect is a magnetotransport phenomenon where a transverse voltage appears in the sample plane when an in-plane magnetic field is applied at an angle to the current.
• It serves as a diagnostic tool for probing band geometry, symmetry, and spin structure in quantum and low-dimensional materials, including twisted bilayer systems.
• Key mechanisms include Berry curvature effects, layer coherence, and strain-induced multipole responses, enabling tunable and even quantized Hall conductance in novel device architectures.

The planar Hall effect (PHE) is a magnetotransport phenomenon in which a transverse voltage appears in the plane of a material when an in-plane magnetic field is applied at an angle to an electric current. Unlike the conventional Hall effect—where the voltage appears orthogonal to both out-of-plane field and current—the PHE is realized entirely within the sample plane and its transverse response vanishes when the field and current are collinear. In topological, quantum, magnetic, and low-dimensional materials, the PHE serves as a sensitive diagnostic of band geometry, symmetry, scattering, spin structure, and topology, and can arise from a range of mechanisms including but not limited to anisotropic magnetoresistance, Berry curvature effects, layer and multipole coherence, spin fluctuations, and orbital magnetoresistance.

1. Fundamental Mechanisms and Symmetry Requirements

The physical origin of the PHE depends critically on both the microscopic properties of the system and its symmetry. In conventional models (e.g., anisotropic ferromagnets), PHE emerges from resistivity anisotropy associated with spin–orbit coupling and broken rotational symmetry. In topological semimetals and insulators, Berry curvature and chiral anomaly mechanisms become prominent. For strictly 2D materials, traditional out-of-plane Berry curvature does not couple to in-plane velocities, rendering the standard PHE forbidden by symmetry unless additional degrees of freedom or symmetry breaking are present (Ghorai et al., 1 May 2024).

In quasi-2D systems (e.g., stacked or twisted van der Waals materials), finite interlayer tunneling hybridizes states across layers, generating hidden planar components of the Berry curvature (Ω^{planar}) and the orbital magnetic moment (m^{planar}) that become active in transport. These components can produce a planar Hall current quadratic in magnetic field, even in the absence of strong intrinsic spin–orbit coupling (Ghorai et al., 1 May 2024).

The general current response is given by

$$j_a = \tau\, \chi_{ab;c}\, E_b\, B_c + \tau\, \chi_{ab;cd}\, E_b\, B_c\, B_d,$$

where the third-rank tensor χ{ab;c} (linear-in-B) tends to be symmetry-forbidden in nonmagnetic 2D systems, making the leading response quadratic-in-B and described by the fourth-rank tensor χ{ab;cd}. The allowed tensor elements are dictated by crystal symmetries such as mirrors, rotations, and time-reversal, as summarized in tables of symmetry restrictions (Ghorai et al., 1 May 2024). In many cases, only certain angular dependencies (e.g., sin 2φ for transverse conductivity) appear, unless additional symmetry breaking, such as strain, is introduced.

2. Layer Coherence and Planar Hall Effect in Van der Waals Bilayers and Trilayers

A mechanism unique to atomically thin bilayer and trilayer materials involves quantum layer coherence—hybridization across layers results in an out-of-plane charge dipole density

$p = -\frac{e d_0}{2} \sigma_z,$

with $d_0$ the interlayer separation and $\sigma_z$ the layer pseudospin. The lateral (in-plane) motion of this dipole generates an in-plane magnetic moment,

$m = \tfrac{1}{2}(p \times v - v \times p),$

that couples to an in-plane magnetic field $\mathbf{B}$ via $-m\cdot \mathbf{B}$ and induces an intrinsic planar Hall response. This effect disappears for layer eigenstates with no coherence, hence only the layer-coherent, hybridized states contribute (Zheng et al., 27 Feb 2024).

Moiré engineering via twist angle, heterostrain, or interlayer sliding allows symmetry tailoring and tunability of the PHE. In twisted bilayer and trilayer systems, small uniaxial strain or sliding can unblock symmetry-forbidden responses and enhance the magnitude and angular structure of the PHE (Zheng et al., 27 Feb 2024).

The general expression capturing the B-linear planar Hall conductivity in this framework is

$$\sigma_H^{(1)} = \frac{e^2}{\hbar} \sum_{n} \int [d k] f_n' \gamma_n(k), \quad \gamma_n(k) = [\hbar v_n \times A_n^{(1)} - \varepsilon_n^{(1)} \Omega_n]_z,$$

where $\varepsilon_n^{(1)} = -m_n \cdot B$ and $A_n^{(1)}$ is the magnetic-field-induced correction to the Berry connection.

The response further generalizes to multipole Hall effects (e.g., dipole Hall current, valley Hall effects induced by in-plane pseudo-magnetic field), demonstrating that quantum layertronics phenomena are extensive in this class of materials (Zheng et al., 27 Feb 2024).

3. Planar Hall Effect from Hidden Planar Band Geometry in Quasi-2D Materials

Hidden geometric band properties—planar Berry curvature and orbital magnetic moment—are activated by finite interlayer tunneling in multilayer van der Waals systems, providing another route to finite 2DPHE (Ghorai et al., 1 May 2024). The band geometry is characterized by

$$\Omega_{n\mathbf{k}}^{\mathrm{planar}} = \frac{2}{\hbar} \mathrm{Re} \sum_{n'\neq n} \frac{\mathbf{v}_{nn'} \times \mathcal{Z}_{n'n}}{\varepsilon_{n\mathbf{k}} - \varepsilon_{n'\mathbf{k}}}$$

and

$$\mathbf{m}_{n\mathbf{k}}^{\mathrm{planar}} = e\, \mathrm{Re} \sum_{n'\neq n} \mathbf{v}_{nn'} \times \mathcal{Z}_{n'n},$$

where $\mathcal{Z}_{n'n}$ is the out-of-plane position operator. The resulting Hall response is quadratic in magnetic field (since T-even tensors govern the effect in absence of magnetic order) and is symmetry-restricted: only certain tensor elements are nonzero for a given crystal class. For example, in Bernal-stacked bilayer graphene, the transverse response is zero unless mirror symmetry is broken, achievable with uniaxial strain (Ghorai et al., 1 May 2024).

The induced current density is then

$j_a = \tau \chi_{ab;cd} E_b B_c B_d,$

and the longitudinal/transverse 2DPHE conductivities follow distinctive angular dependences, e.g.,

$$\sigma_{\perp} = \tau B^2 (\chi_{yx;xx} \cos^2\phi + \chi_{yx;yy} \sin^2\phi + \chi_{yx;xy} \sin\phi\cos\phi).$$

The 2DPHE is sensitive to gate-induced Lifshitz transitions and to the carrier density, and its magnitude can be enhanced by increasing the number of layers (Ghorai et al., 1 May 2024).

4. Quantized Planar Hall States and Topological Phase Transitions

Beyond linear transport, strong in-plane magnetic fields in twisted trilayer systems can drive band inversion and topological phase transitions accompanied by quantized Hall conductance. In twisted trilayer MoTe₂, an interlayer bias combined with sufficiently strong in-plane magnetic field changes the Chern number of spin-resolved bands, leading to a transition from nonquantized to quantized PHE (Zheng et al., 27 Feb 2024). This scenario is distinct from conventional quantum Hall or QAHE states that rely on out-of-plane fields or spin-orbit coupling: here, quantization emerges from layer coherence and band geometry, opening an avenue for topological engineering of Hall responses through in-plane fields and interlayer design.

5. Multipole and Valley Planar Hall Effects

The flexibility of layered van der Waals materials allows generalization of the PHE concept from monopole charge currents to higher-order moments (multipole PHE) and even valley Hall effects. When strain is present, it can act as an in-plane pseudomagnetic field with opposite sign at different valleys, enabling a valley-resolved planar Hall response—an effect not accessible in centrosymmetric systems without such symmetry breaking (Zheng et al., 27 Feb 2024).

The distinction between monopole (total charge current), dipole (current difference between top and bottom layers), and valley (valley-contrasted) planar Hall currents provides a broader conceptual framework for understanding transport phenomena in layered and moiré superlattices.

6. Implications for Materials Physics and Quantum Layertronics

These advances transform the understanding of planar Hall transport in atomically thin van der Waals structures. The intrinsic coupling of layer, valley, and multipole degrees of freedom to in-plane fields enables new device functionalities:

• Gate- and strain-tunable Hall sensors and logic elements.
• Manipulation and detection of valley or layer-polarized currents for valleytronic and layertronic applications.
• Potential realization of quantized Hall states in flat-band moiré materials via in-plane field engineering.
• Exploitation of enhanced density of states and Berry curvature in moiré flat bands for high-efficiency switching.

Because the discussed mechanisms operate independently of traditional 3D orbital motion or strong spin–orbit coupling, they are particularly relevant for low-dimensional and weak-SOC materials, greatly expanding the platform for quantum information and topological electronics.

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Table: Mechanistic Classification of Planar Hall Effects in (Quasi-)2D Materials

Mechanism	Key Ingredient	Example Material/Class
Layer coherence	Hybridized dipole moment	Twisted bilayer/trilayer
Hidden planar Berry curvature/OMM	Interlayer tunneling	Bilayer graphene, vdW stacks
Quantized PHE via topological transition	Band inversion under B	Twisted trilayer MoTe₂
Multipole/valley planar Hall effects	Strain/pseudomagnetic field	Strained TBG, MoTe₂

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The new discoveries of layer coherence and hidden planar band-geometry-driven PHE in atomically thin quantum materials not only fill the gap left by conventional paradigms, but also establish flexible platforms for controlling and quantizing Hall responses and for advancing quantum layertronics and moiré-based device physics (Zheng et al., 27 Feb 2024, Ghorai et al., 1 May 2024).