Universal classes of disorder scatterings in in-plane anomalous Hall effect

This paper theoretically investigates the universal classes of disorder scattering (scalar, spin-conserving, and spin-flipping) in the in-plane anomalous Hall effect using a massive Dirac fermion model, revealing distinct disorder-dependent conductivity behaviors and novel sinusoidal oscillations arising specifically from spin-flipping scattering.

The Gist

Imagine you are trying to drive a car through a city. In a perfect, empty city with no traffic, your car would move in a straight line, and its path would be determined entirely by the road's design (the "intrinsic" nature of the road).

Now, imagine that city is full of obstacles: potholes, construction zones, and other drivers who might cut you off or push your car sideways. These are the disorders (impurities and defects) in a material.

This paper is about a specific type of traffic jam in the world of quantum physics called the In-Plane Anomalous Hall Effect (IPAHE).

The Big Picture: What is IPAHE?

Usually, when you push electricity through a magnetic material, the electrons get pushed sideways, creating a voltage at a 90-degree angle. This is the standard "Hall Effect."

But in IPAHE, the electrons are pushed sideways within the same flat plane as the magnetic field. Think of it like driving on a flat highway where, instead of turning left or right, your car mysteriously drifts sideways along the lane, even though you aren't turning the steering wheel. Scientists are very interested in this because it could lead to super-efficient, low-energy electronics.

For a long time, physicists thought this sideways drift was mostly caused by the "shape" of the energy roads (the intrinsic part). They assumed the messy obstacles (disorder) didn't matter much.

This paper says: "Wait a minute! The obstacles matter a lot, and they behave in three very different ways."

The Three Types of "Traffic Obstacles"

The authors looked at three specific types of "disorder" (impurities) that electrons bump into. They used a creative model of a "warping" road (a hexagonal pattern) to see how these obstacles affect the drift.

1. The "Bump in the Road" (Scalar Scattering)

• The Analogy: Imagine a pothole or a speed bump. It's just a physical obstacle. It doesn't care which way your car is facing or what color it is.
• The Physics: These are non-magnetic impurities. They slow electrons down but don't flip their internal "spin" (a quantum property like a tiny compass needle).
• The Result: The sideways drift (Hall effect) behaves in a predictable, standard way. It's like hitting a bump; you slow down, but you don't suddenly start spinning.

2. The "Magnetic Gatekeeper" (Spin-Conserving Scattering)

• The Analogy: Imagine a gatekeeper who only lets cars with a specific compass direction (North) pass through, but blocks those pointing South. However, if your compass is already pointing North, they let you through without changing your direction.
• The Physics: These are magnetic impurities that interact with the electron's spin but keep the "up/down" direction the same.
• The Result: The drift changes slightly compared to the first type, but it still follows the general rules of the road. It's a bit more complex, but the pattern remains familiar.

3. The "Spin-Flipper" (Spin-Flipping Scattering)

• The Analogy: This is the wild card. Imagine a mischievous traffic cop who, every time you hit a bump, spins your car 180 degrees or flips your compass needle from North to South.
• The Physics: These are magnetic impurities that actively flip the electron's spin.
• The Result: This is the paper's biggest discovery. When electrons hit these "spin-flippers," the sideways drift doesn't just happen; it starts oscillating (wiggling) in a very specific, rhythmic pattern.
  • Instead of a smooth drift, the effect wiggles with a period of $\pi$ (180 degrees) and $2\pi$ (360 degrees).
  • Why it matters: Standard physics predicts a smooth, three-fold pattern (like a Mercedes logo). But with these spin-flippers, the pattern breaks symmetry and starts looking like a sine wave. It's like the car suddenly starts swaying left-right-left-right as it drives, creating a brand new kind of signal.

Why Should We Care?

The authors found that by understanding these three types of "traffic," we can explain why some experiments in real materials (like topological insulators) look weird or don't match the old theories.

• The "Warping" Factor: The material they studied has a "hexagonal warping" (the road is shaped like a hexagon, not a circle). This shape, combined with the "spin-flipping" obstacles, creates those unique, rhythmic wiggles in the electrical current.
• The Takeaway: If you want to build better, low-energy electronic devices, you can't just ignore the dirt and dust (disorder) in your materials. In fact, by controlling the type of disorder (specifically the spin-flipping kind), you might be able to tune the electrical current to do exactly what you want.

Summary in One Sentence

This paper reveals that the "messiness" of a material isn't just noise; depending on whether the impurities flip the electron's spin or not, they can create entirely new, rhythmic patterns in electrical flow that we can use to build smarter, more efficient technology.

Technical Summary

1. Problem Statement

The In-Plane Anomalous Hall Effect (IPAHE) is a transport phenomenon where the Hall current and magnetization (or magnetic field) lie within the same plane. While recent experimental progress has highlighted its potential for low-energy electronics, theoretical understanding remains incomplete.

• Current Gap: Existing research primarily focuses on the intrinsic contribution to IPAHE, which arises from the Berry curvature of electronic bands.
• The Missing Link: The extrinsic contributions, driven by disorder scattering (specifically skew scattering and side jump), are largely unexplored for IPAHE.
• Specific Challenge: Real magnetic materials contain various types of disorder (scalar/non-magnetic, spin-conserving magnetic, and spin-flipping magnetic). It is unclear how these distinct "universality classes" of disorder affect the extrinsic IPAHE, particularly in systems with complex band structures like those with hexagonal warping.

2. Methodology

The authors employ a rigorous theoretical framework based on the Kubo formula within the non-crossing approximation (NCA) (ladder approximation).

• Model System: They utilize a prototypical two-dimensional (2D) massive Dirac fermion model incorporating a hexagonal warping term. This model describes the surface states of topological insulators under generic Zeeman fields (both in-plane $M_{\parallel}$ and out-of-plane $M_z$).
  • Hamiltonian: $H_0(\mathbf{k}) = v(k_y\sigma_x - k_x\sigma_y) + \lambda k \sigma_z + \mathbf{M} \cdot \hat{\sigma}$, where $\lambda k$ represents the hexagonal warping.
• Disorder Classification: The study systematically investigates three universal classes of disorder scattering:
  1. Class A (Scalar): Non-magnetic impurities ($V_0 \hat{\sigma}_0$).
  2. Class B (Spin-Conserving): Magnetic impurities conserving the $z$-component of spin ($V_z \hat{\sigma}_z$).
  3. Class C (Spin-Flipping): Magnetic impurities causing spin flips via in-plane random magnetic moments ($V_x \hat{\sigma}_x, V_y \hat{\sigma}_y$).
• Calculation Techniques:
  • Self-Energy & Vertex Corrections: Calculated using the Dyson equation and ladder diagrams to determine the renormalized Green's functions and velocity operators.
  • Higher-Order Correlations: To capture skew scattering, the authors include third-order non-Gaussian disorder correlations.
  • Crossed Diagrams: They evaluate "X" and "$\Psi$" diagrams (crossed impurity lines) to ensure the robustness of the angular dependence results.

3. Key Contributions

• Systematic Classification: The paper provides the first comprehensive theoretical derivation of extrinsic IPAHE contributions for all three universal disorder classes in a warped Dirac system.
• Symmetry Breaking Analysis: It elucidates how different disorder types break or preserve crystal symmetries (specifically $C_{3v}$), leading to distinct angular dependencies in the Hall conductivity.
• Novel Oscillatory Behavior: The discovery that spin-flipping scattering (Class C) induces nontrivial sinusoidal oscillations with periods of $\pi$ and $2\pi$ in the extrinsic Hall conductivity, a feature absent in standard isotropic massive Dirac fermions.
• Magnetoresistance Mechanism: The derivation of in-plane magnetoresistance (Planar Hall Effect) mechanisms driven by hexagonal warping, offering an alternative explanation for experimental anomalies in topological insulators.

4. Key Results

A. Intrinsic vs. Extrinsic Conductivity

• Intrinsic Part ($\sigma^{in}_{xy}$): Depends on $M_z$ and the warping parameter $\lambda$. It exhibits a $3\theta$ dependence (period $2\pi/3$) due to the $C_{3v}$ symmetry of the warping term.
• Side Jump ($\sigma^{sj}_{xy}$):
  • Independent of disorder density and strength (similar to intrinsic AHE).
  • Classes A & B: Preserve $C_{3v}$ symmetry; contributions depend on $M_z$ and $M_{\parallel}$ with specific angular forms.
  • Class C: Breaks $C_{3v}$ symmetry, introducing terms proportional to $M_{\parallel} \sin \theta$.

B. Skew Scattering ($\sigma^{sk}_{xy}$)

• Dependence: Proportional to disorder density ($n_i^{-1}$) and scattering strength parameters ($u_1^3/u_0^4$).
• Class A & B: Respect $C_{3v}$ symmetry, showing $3\theta$ dependence.
• Class C (Spin-Flipping):
  • Exhibits quadratic dependence on in-plane magnetization ($M_{\parallel}^2$).
  • Shows $\pi$ and $2\pi$ periodicity (terms like $\sin 2\theta$ and $\cos 2\theta$).
  • This is a distinct signature of spin-flipping scattering, contrasting with the standard massive Dirac fermion results where skew scattering vanishes or follows different symmetries.

C. Total Anomalous Hall Conductivity

The total conductivity $\sigma_{xy} = \sigma^{in} + \sigma^{sj} + \sigma^{sk}$ reveals that:

• In the presence of spin-flipping impurities, the extrinsic part can dominate or significantly modify the intrinsic signal.
• The angular dependence of the Hall signal becomes highly sensitive to the relative weights of different in-plane impurity components.

D. In-Plane Magnetoresistance (Planar Hall Effect)

• The authors calculated the symmetric part of the conductivity tensor ($\sigma^S_{xy}$).
• They found that hexagonal warping induces anisotropic magnetoresistance with fourfold symmetry (superposition of $\pi$ and $\pi/2$ periods).
• This provides a theoretical mechanism to explain deviations from the conventional $\sin 2\theta$ oscillation observed in experiments on Sn-doped topological insulators ($Bi_{1.1}Sb_{0.9}Te_2S$).

E. Crossed Diagrams

• The inclusion of crossed impurity diagrams (X and $\Psi$) results in a substantial correction to the amplitude of the conductivity (approx. $\pm 0.8$ times the non-crossing contribution) but does not alter the angular dependence or the periodicity of the IPAHE.

5. Significance

• Theoretical Framework: This work fills a critical gap in the theory of anomalous transport by rigorously defining how different disorder universality classes modify the IPAHE.
• Experimental Guidance: The prediction of $\pi$ and $2\pi$ periodic oscillations in the extrinsic Hall signal serves as a specific signature for identifying spin-flipping scattering mechanisms in experiments.
• Material Design: Understanding the interplay between warping, magnetization, and disorder types aids in the design of energy-efficient spintronic devices and the interpretation of magnetotransport data in topological materials and kagome metals.
• Resolution of Anomalies: The proposed mechanism for fourfold symmetric magnetoresistance offers a solution to previously unexplained features in topological insulator experiments.

In conclusion, the paper demonstrates that disorder scattering is not merely a perturbation but a fundamental factor that can qualitatively change the symmetry and periodicity of the in-plane anomalous Hall effect, particularly when spin-flipping processes are involved.