🔬 mesoscale physics

Disorder-induced modulation of the nonlinear Hall effect in Weyl semimetals

This paper investigates how different types of impurity scattering, particularly polarized magnetic impurities, modulate the nonlinear Hall effect in Weyl semimetals by deriving conductivity tensors within a semiclassical Boltzmann framework and revealing that while magnetic disorder introduces a helicity-dependent anisotropic correction, the isotropic response from scalar disorder remains dominant.

Original authors: Juan A. Cañas, Daniel A. Bonilla, A. Martín-Ruiz

Published 2026-09-18✓ Author reviewed ⓘ

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CC BY 4.0

Original authors: Juan A. Cañas, Daniel A. Bonilla, A. Martín-Ruiz

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of modern physics, there exists a class of materials known as topological semimetals. These are not your ordinary solids; their internal electronic structure possesses a unique, unbreakable geometry that gives rise to strange and robust behaviors. Imagine a material where electrons move as if they have no mass, zipping through the crystal lattice at incredible speeds. In a specific type called a Weyl semimetal, these electrons behave like tiny magnets with a distinct handedness, or "chirality." They come in two varieties, left-handed and right-handed, which act as sources and sinks for a hidden geometric field called the Berry curvature. This field acts like a subtle, invisible wind that pushes the electrons sideways when they are pushed by an electric field, creating a phenomenon known as the Hall effect. While scientists have long understood how these materials respond to steady, gentle pushes, a new frontier has opened up: what happens when the push is rapid, oscillating, and strong enough to trigger nonlinear responses? Understanding this is crucial because these materials could one day power ultra-fast electronic devices or serve as the foundation for new types of optical sensors, but only if we can predict how they behave when real-world imperfections are present.

Real materials are never perfect. They contain impurities—tiny defects or foreign atoms scattered throughout the crystal—that disrupt the smooth flow of electrons. For decades, physicists have struggled to predict exactly how these messy imperfections interact with the exotic geometry of Weyl semimetals, especially when the material is subjected to the intense, rapidly changing electric fields of light. A team of researchers from the National Autonomous University of Mexico has now mapped out this complex relationship with remarkable precision. They set out to answer a fundamental question: how do different types of impurities change the way these materials generate electrical currents when hit with oscillating light? By building a detailed theoretical model, they calculated how electrons scatter off various kinds of disorder and how this scattering alters the material's ability to produce a nonlinear Hall effect, a phenomenon where the electrical response is not just a simple copy of the input but contains new frequencies and rectified currents.

The researchers focused on two main categories of impurities. The first group consists of "scalar" impurities, which are like neutral, non-magnetic obstacles that simply block the path of an electron without caring about its internal spin or handedness. They modeled these as short-range bumps, Gaussian clouds, and screened electric charges. Their calculations revealed a clear rule: for these neutral obstacles, the scattering process is the same for both left-handed and right-handed electrons. Because the two types of electrons in a Weyl semimetal naturally produce opposing currents that tend to cancel each other out, a net signal only appears if the two types of electrons are not perfectly identical in energy. If the material is perfectly symmetric, the currents cancel to zero. However, if the energy levels of the two electron types are slightly different, the cancellation is incomplete, and a measurable current emerges. This current is driven entirely by the geometric "wind" of the Berry curvature, and its strength depends on the specific energy of the electrons and the type of impurity present.

The second group of impurities is far more intriguing: polarized magnetic impurities. These are atoms with a fixed magnetic orientation, acting like tiny, aligned bar magnets embedded in the crystal. The researchers found that these magnetic obstacles behave very differently from their neutral counterparts. Because the electrons in a Weyl semimetal have their spin locked to their direction of motion, a magnetic impurity interacts with them in a way that depends on the electron's specific path and handedness. Through a complex interplay of quantum mechanical effects, these magnetic impurities create an "anisotropic" scattering rate, meaning the electrons scatter differently depending on their direction relative to the magnetic alignment. This breaks the symmetry of the system in a new way, allowing the material to generate a second-harmonic current—a signal that oscillates at twice the frequency of the incoming light. This is a significant finding because, in a perfectly symmetric material without these magnetic impurities, such a signal would be strictly forbidden.

However, the study also delivered a sobering reality check regarding the size of these effects. While the magnetic impurities do theoretically allow for this new type of current and change the mathematical structure of the response, the researchers calculated that the magnitude of this anisotropic effect is incredibly small. For the realistic parameters they used, the new magnetic contribution is three to four orders of magnitude weaker than the dominant, isotropic response caused by the standard geometric effects. In other words, while the magnetic impurities open a new door in the physics of these materials, the room behind it is very quiet compared to the noise of the main hall. The dominant signal still comes from the geometric properties of the electrons and the slight energy differences between the two types of nodes, rather than the magnetic alignment itself.

The work provides a comprehensive microscopic blueprint for how disorder shapes the nonlinear behavior of these exotic materials. By separating the geometric properties of the electrons from the messy details of impurity scattering, the researchers showed that the transport relaxation time—the average time an electron travels before hitting a defect—is the key variable that connects the microscopic world of impurities to the macroscopic world of electrical currents. Their findings confirm that to see a strong nonlinear Hall effect in a Weyl semimetal, one needs a material where the two types of electron nodes are energetically distinct, and they clarify that while magnetic impurities can introduce new directional dependencies, they are unlikely to be the primary driver of large signals in standard conditions. This clarity is essential for experimentalists who are currently trying to measure these effects in the lab, as it tells them exactly what to look for and what to expect, ensuring that future experiments are guided by a solid understanding of how the invisible geometry of the crystal interacts with the inevitable imperfections of the real world.

Technical Summary: Disorder-Induced Modulation of the Nonlinear Hall Effect in Weyl Semimetals

Problem Statement
Topological semimetals, particularly Weyl semimetals, exhibit rich nonlinear electronic and optical phenomena driven by their nontrivial band geometry and Berry curvature. While linear transport properties are well-established, the mechanisms governing nonlinear Hall responses (specifically rectified DC and second-harmonic AC currents) in the presence of disorder remain less understood. A critical open question is how the microscopic details of impurity scattering—specifically the momentum dependence of the transport relaxation time and the nature of the disorder (scalar vs. magnetic)—interact with the symmetry breaking of the Weyl nodes to determine the magnitude and tensor structure of the nonlinear Hall conductivity. Previous theoretical works have often treated disorder phenomenologically or focused on extrinsic mechanisms like skew-scattering and side-jump; this work aims to rigorously derive how explicit microscopic scattering potentials modulate the nonlinear response within a semiclassical framework.

Methodology
The authors employ a semiclassical Boltzmann approach to describe the nonlinear Hall response of a two-node Weyl semimetal model. The methodology proceeds in three main stages:

Semiclassical Formalism: The authors derive general expressions for the rectified and second-harmonic conductivity tensors up to second order in the external electric field. Unlike approaches that introduce anomalous driving terms into the equations of motion, this work captures disorder effects strictly through a general, momentum-dependent transport relaxation time, $\tau_{\alpha\mathbf{k}}$ . The charge current is decomposed into a quasiparticle contribution (driven by the group velocity and distribution function correction) and an anomalous contribution (driven by the Berry curvature and the first-order distribution correction).
Microscopic Scattering Models: The transport relaxation time is evaluated microscopically for four distinct impurity mechanisms using the Born approximation and T-matrix expansions:
- Scalar Disorder: Short-range (delta), finite-range Gaussian, and screened Coulomb potentials.
- Magnetic Disorder: Localized exchange interactions with both unpolarized and polarized magnetic moments.
  For magnetic impurities, the authors go beyond the first Born approximation, analyzing the interference between first- and second-order scattering amplitudes to determine if helicity-dependent relaxation rates emerge.
Evaluation of Conductivities: The derived relaxation times are substituted into the general conductivity formulas. The authors analyze the angular integrals over the Fermi surface to determine how isotropic versus anisotropic relaxation times affect the tensor structure of the nonlinear Hall response.

Key Contributions and Results

Separation of Geometric and Microscopic Effects: The work establishes a unified framework where the nonlinear Hall response is determined by two distinct ingredients: the geometric properties of the electronic structure (band velocity and Berry curvature) and the microscopic scattering details encoded in $\tau_{\alpha\mathbf{k}}$ .
Scalar Disorder: For scalar impurities (delta, Gaussian, screened Coulomb), the transport relaxation time is found to be isotropic and independent of the Weyl node chirality. Consequently, the quasiparticle contribution to the nonlinear Hall current vanishes due to rotational symmetry. The net response is entirely governed by the anomalous velocity associated with the Berry curvature. A finite net response requires the two Weyl nodes to be energetically inequivalent (i.e., displaced in energy, $b_{0+} \neq b_{0-}$ ) to prevent the exact cancellation of contributions from opposite chiralities. The magnitude of the response depends on the specific energy dependence of the relaxation time for each impurity model.
Magnetic Disorder and Anisotropy:
- Unpolarized Impurities: Even with spin-momentum locking, unpolarized magnetic impurities do not generate different transport lifetimes for opposite helicities at the level of the first Born approximation or the pure second-order term.
- Polarized Impurities: A finite polarization of magnetic moments introduces a preferred direction. The authors find that the interference between first- and second-order Born amplitudes generates a helicity-dependent, anisotropic correction to the relaxation time. This correction depends on the projection of the incoming momentum onto the impurity polarization axis ( $\hat{m} \cdot \hat{k}$ ).
- Tensor Structure: This anisotropy modifies the tensor structure of the nonlinear Hall conductivity, promoting the scalar response function $\psi(\omega)$ into two independent components: longitudinal ( $\psi_{\parallel}$ ) and transverse ( $\psi_{\perp}$ ) relative to the impurity polarization.
Second-Harmonic Generation: In the isotropic scalar case, the second-harmonic current vanishes for linearly polarized light and is zero for the anomalous contribution in the symmetric two-node model. However, polarized magnetic impurities generate a finite anomalous second-harmonic current for specific orientations of the electric field relative to the impurity polarization, even if the underlying Weyl dispersion remains isotropic.
Magnitude of Effects: While the anisotropic contribution from polarized magnetic impurities is theoretically significant for altering the tensor structure, numerical evaluations for representative parameters (based on materials like EuCd $_2$ As $_2$ ) show that the anisotropic correction is several orders of magnitude smaller than the dominant isotropic response.

Significance and Claims
The paper claims to provide a rigorous microscopic understanding of how impurity scattering enters the nonlinear Hall response of Weyl semimetals. Its primary significance lies in demonstrating that:

The nonlinear Hall response is not solely determined by band topology but is critically modulated by the microscopic form of the transport relaxation time.
For scalar disorder, the requirement for energetic inequivalence of Weyl nodes is essential to observe a finite nonlinear Hall effect due to chirality cancellation.
Polarized magnetic impurities introduce a unique mechanism for chirality-dependent momentum relaxation via interference terms, leading to anisotropic corrections and enabling second-harmonic generation in regimes where it would otherwise vanish.
Despite the qualitative differences introduced by magnetic disorder, the quantitative impact of the anisotropic terms is small compared to the isotropic background for typical material parameters.

The authors conclude that their semiclassical formalism, which separates geometric contributions from scattering details, offers a flexible foundation for future studies of nonlinear transport in other topological systems, including tilted cones, Dirac semimetals, and nodal-line systems, as well as for incorporating extrinsic mechanisms like side-jump and skew-scattering.

Technical Summary: Disorder-Induced Modulation of the Nonlinear Hall Effect in Weyl Semimetals

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