Phase-Shifted Planar Hall and Magnetoresistive Responses in Weyl Semimetals

Abstract: The planar Hall resistivity and magnetoresistivity of Weyl semimetals are conventionally expected to exhibit $\sin 2φ$ and $\cos2φ$ angular dependences, respectively, where $φ$ is the angle between electric and magnetic fields. However, experiments reveal shifted extrema in the planar Hall signal and sign reversals in magnetoresistivity at $φ< π/4$. Here, using a diagrammatic Kubo-formula approach, we identify an intrinsic quadratic magnetic-field contribution to the planar transport response that is absent in conventional semiclassical description. This contribution introduces an additional term proportional to $\cos 2 φ$ and $\sin 2φ$, respectively, in transverse and longitudinal conductivities. Consequently, both planar Hall and magnetoresistive responses acquire a phase-shifted form $\sin 2(φ+φ_r)$ and $\cos 2(φ+φ_r)$, respectively. The same phase shift extracted independently from longitudinal and transverse responses quantitatively describes available experimental data. Our results establish a microscopic origin of the anomalous angular dependence observed in Weyl semimetals.

• The paper establishes that intrinsic quantum quadratic contributions, captured by the Kubo formalism, generate a finite phase shift in magnetotransport responses.
• The angular dependencies in planar Hall and longitudinal resistivities are expressed as sinusoidal functions with a measurable phase shift, reconciling experimental discrepancies.
• Phase shift magnitude correlates with disorder levels and chemical potential, underscoring its utility in probing intrinsic quantum transport in Weyl semimetals.

Phase-Shifted Magnetotransport in Weyl Semimetals: Microscopic Origins and Quantum Response

Introduction

The study investigates the angular magnetotransport characteristics in Weyl semimetals (WSMs), focusing on the planar Hall effect (PHE) and anisotropic magnetoresistivity (AMR). Conventionally, semiclassical Boltzmann transport theory predicts $\sin 2\phi$ and $\cos 2\phi$ angular dependencies for planar Hall and longitudinal resistivities, respectively, with $\phi$ being the in-plane angle between the applied electric and magnetic fields. However, experimental data consistently report shifted extrema in the PHE and sign inversions in magnetoresistivity at angles $\phi < \pi/4$, which are not captured by semiclassical theory. The paper addresses this discrepancy through a Kubo-formalism-based quantum transport calculation, uncovering an intrinsic phase shift attributable to Fermi-sea contributions and symmetry-allowed quadratic magnetic responses.

Symmetry-Allowed Quadratic Response and the Phase Shift

In the presence of Weyl-node separation breaking rotational symmetry, the charge current in response to in-plane electric and magnetic fields admits terms beyond $(\mathbf{E} \cdot \mathbf{B}) \mathbf{B}$, specifically those involving $$(\mathbf{E} \cdot (\hat{z} \times \mathbf{B}))\mathbf{B}$$, $$(\mathbf{E} \cdot \mathbf{B})(\hat{z} \times \mathbf{B})$$, and $$(\mathbf{E} \cdot (\hat{z} \times \mathbf{B})) (\hat{z} \times \mathbf{B})$$. These additional symmetry-allowed quadratic terms contribute to the planar Hall ($\sigma_{yx}$) and longitudinal magnetoconductivity ($\sigma_{xx}$), generating an angular dependency that can be compactly formulated as:

$\cos 2\phi$0

$\cos 2\phi$1

where $\cos 2\phi$2 acts as the phase shift. This directly explains the shifted extrema and zero-crossing in the angular dependence observed experimentally.

Quantum Transport Calculation: Kubo Formalism

The derivation proceeds via diagrammatic Kubo formalism, encompassing both Fermi-surface and Fermi-sea contributions, unlike the semiclassical approach which captures only the former. The underlying minimal Hamiltonian incorporates chiral Weyl nodes separated along the $\cos 2\phi$3-axis, with the magnetic response expanded to second order in magnetic field using vector potential in a suitable gauge.

The leading quadratic magnetic response arises from the second-order diagrams with two magnetic vertices on opposite fermion lines. The resultant conductivities contain both disorder-dependent ($\cos 2\phi$4) and disorder-independent ($\cos 2\phi$5) contributions:

• $\cos 2\phi$6: scales with relaxation time $\cos 2\phi$7, dominant in conventional Boltzmann theory (Fermi-surface), includes higher-order $\cos 2\phi$8 dependence from quantum response.
• $\cos 2\phi$9: independent of $\phi$0, stemming from intrinsic quantum (Fermi-sea) effects, robust in the clean limit.

Explicit evaluation of the response functions (see end-matter for technical calculations) yields the dependencies for $\phi$1 and $\phi$2, with $\phi$3 present only when the chemical potential is above the Weyl nodes ($\phi$4), reinforcing its intrinsic origin.

Experimental Validation and Parameter Extraction

The phase-shifted angular dependencies were fitted to digitized experimental data from several WSM systems, including Cd$\phi$5As$\phi$6 and trigonal-PtBi$\phi$7, using

$\phi$8

$\phi$9

The extracted phase shifts ($\phi < \pi/4$0) and amplitude parameters ($\phi < \pi/4$1) from both longitudinal and transverse responses are quantitatively consistent for each sample and field strength. Notably, PtBi$\phi < \pi/4$2 exhibits a larger phase shift, suggesting a stronger intrinsic contribution compared to Cd$\phi < \pi/4$3As$\phi < \pi/4$4.

Numerical estimations utilizing experimentally relevant parameters for $\phi < \pi/4$5, $\phi < \pi/4$6, $\phi < \pi/4$7, $\phi < \pi/4$8, and the cutoff $\phi < \pi/4$9 reproduce the observed phase shift. As disorder increases (i.e., as $(\mathbf{E} \cdot \mathbf{B}) \mathbf{B}$0 decreases), $(\mathbf{E} \cdot \mathbf{B}) \mathbf{B}$1 increases, reflecting the relative change between intrinsic and extrinsic transport channels.

Implications and Future Directions

The results establish the microscopic quantum origin of anomalous angular dependencies in planar Hall and magnetoresistive responses of WSMs, highlighting the necessity to go beyond semiclassical transport theory and account for Fermi-sea contributions via full quantum mechanical treatment. Practically, the extracted phase shift serves as a probe of intrinsic quantum magnetotransport, offering new avenues for characterizing WSMs and investigating topological quantum materials.

Theoretically, the presence of $(\mathbf{E} \cdot \mathbf{B}) \mathbf{B}$2-independent contributions implies the persistence of the phase shift in ultraclean samples, and its weak dependence on magnetic field amplitude suggests minimal susceptibility to higher-order magnetic corrections. The robustness of intrinsic quantum responses against disorder positions WSMs as platforms for further exploring topological magnetotransport and for device applications relying on angular-sensitive quantum response.

Open directions include a more detailed microscopic analysis of magnetic-field-induced modifications to electronic wavefunctions and geometric properties, especially near the Weyl nodes, as well as exploration of quantum geometric contributions and their interplay with strong correlation effects in multi-orbital WSM systems.

Conclusion

The paper demonstrates that Weyl semimetals' planar Hall effect and magnetoresistive responses exhibit a finite phase shift due to intrinsic quantum quadratic magnetic-field contributions. The angular dependencies $(\mathbf{E} \cdot \mathbf{B}) \mathbf{B}$3 and $(\mathbf{E} \cdot \mathbf{B}) \mathbf{B}$4 reconcile longstanding experimental discrepancies and reveal the essential role of symmetry, Fermi-sea quantum response, and Berry curvature in WSM magnetotransport. These insights enhance the fundamental understanding of topological transport phenomena and underpin advances in both materials characterization and quantum device engineering.