Disorder induced time-reversal-odd nonlinear spin and orbital Hall effects

This paper develops a comprehensive theory for disorder-induced, time-reversal-odd nonlinear spin and orbital Hall effects, revealing that mechanisms such as coordinate shifts, side-jumps, and skew scattering can generate significant orbital angular-momentum currents that often surpass their spin counterparts.

The Gist

Imagine a bustling city where electrons are the commuters. Usually, when we talk about electricity, we care about how these commuters move in a straight line (current). But in the world of quantum physics, these commuters also have a "spin" (like a spinning top) and an "orbit" (like a planet circling a sun). Together, these create angular momentum.

For decades, scientists have studied how these spinning commuters get deflected sideways when they hit obstacles (disorder), creating a "Hall effect." This is usually a linear process: push them a little, they move a little to the side.

This paper is about a new, more chaotic traffic jam.

The researchers are looking at what happens when you push these electrons really hard (nonlinear transport) and how "disorder" (impurities, defects, and bumps in the road) actually helps create a sideways flow of angular momentum, rather than just stopping it.

Here is the breakdown of their discovery using simple analogies:

1. The Two Types of Commuters: Spin vs. Orbit

Think of the electrons as having two distinct personalities:

• The Spin Commuter: This is the electron's intrinsic spin. It's like a person spinning on their own axis while walking.
• The Orbit Commuter: This is the electron's path around the atom's nucleus. It's like a person walking in a large circle around a park.

For a long time, scientists thought the "Spin" commuter was the only one who mattered for generating currents. This paper says, "Wait a minute! The 'Orbit' commuter is just as important, and in some cases, much stronger."

2. The "Disorder" is the Hero, Not the Villain

In a perfect, crystal-clear city, electrons move smoothly. But real materials are messy. They have potholes, construction zones, and random obstacles (disorder).

Usually, we think disorder just slows traffic down. But this paper reveals that in a nonlinear world (where the push is strong), disorder actually creates new ways for the "Orbit" and "Spin" commuters to get pushed sideways.

The authors identified four specific "disorder tricks" that cause this:

• The Side-Jump: Imagine a commuter trying to dodge a pothole. They don't just stop; they take a sudden, tiny leap to the side. This "leap" creates a current.
• The Skew Scattering: Imagine a billiard ball hitting a rough patch of cloth. Instead of bouncing straight back, it gets deflected at a weird angle. This "skew" bounce pushes the commuters sideways.
• The Coordinate Shift: When an electron hits a bump, its "address" (position) shifts slightly before it even moves. It's like a car getting pushed sideways just by hitting a bump, even before the driver steers.
• The Anomalous Scattering: A weird, quantum-mechanical glitch where the electron scatters in a way that defies normal expectations, creating a sideways flow.

3. The Big Surprise: The "Orbit" Wins

The researchers built a mathematical model (a simulation of a specific type of material) to see which "commuter" was stronger.

The Result: In many scenarios, especially in materials with a small energy gap (like a small hill the electrons have to climb), the Orbital current is not just equal to the Spin current—it can be much, much larger.

Analogy: Imagine a race between a spinning top (Spin) and a planet (Orbit). In a smooth race, the top wins. But in a bumpy, chaotic race with lots of obstacles, the planet's massive size and momentum allow it to dominate the track, pushing the top aside.

4. Why This Matters: The "Recipe" for Future Tech

The paper doesn't just say "it happens." It gives scientists a recipe (a scaling law) to figure out which disorder trick is happening in a real experiment.

• The Problem: If you measure a current in a lab, you see a number. You don't know if it came from the "Side-Jump" or the "Skew Scatter."
• The Solution: The authors say, "Change the temperature and the amount of dirt (disorder) in your material. If the current changes in this specific pattern, it's the Side-Jump. If it changes in that pattern, it's Skew Scattering."

This is like a detective using a specific type of fingerprint to identify which criminal broke into the house.

The Takeaway

This paper opens a new door in "Spintronics" (electronics based on spin). It tells us:

1. Don't ignore the Orbit: The orbital motion of electrons is a massive, untapped resource for generating currents.
2. Embrace the Mess: Disorder isn't just a nuisance; it's a tool we can use to generate powerful new currents.
3. New Materials: We should look for materials with specific symmetries (like certain magnetic crystals) where these "Time-Reversal-Odd" effects are strongest.

In short, the authors have found a new way to harness the chaos of the quantum world to create more efficient ways to move information and energy, proving that sometimes, a little bit of disorder is exactly what you need to get things moving.

Technical Summary

1. Problem Statement

While the role of disorder scattering in linear Hall transport (both spin and orbital) is well-established, a comprehensive understanding of disorder-induced mechanisms in nonlinear angular-momentum transport is lacking. Specifically:

• Previous studies on nonlinear transport (e.g., nonlinear Hall effect, nonlinear Edelstein effect) have identified various extrinsic mechanisms (side-jump, skew scattering, coordinate shift, anomalous scattering amplitude).
• However, a unified theoretical framework describing time-reversal-odd ($T$-odd) second-order angular-momentum currents (both spin and orbital) that incorporates these disorder mechanisms is missing.
• There is a need to distinguish between intrinsic contributions (Berry curvature dipoles) and extrinsic disorder-induced contributions, and to understand the relative magnitude of orbital versus spin components in the nonlinear regime.

2. Methodology

The authors developed a semiclassical theory based on the Boltzmann transport equation, treating spin and orbital angular momentum on an equal footing.

• Formalism:
  • Defined the angular-momentum current operator $\hat{j}^{\alpha\beta}_X$ involving both orbital ($\hat{L}$) and spin ($\hat{S}$) components.
  • Expanded the distribution function $f_l$ and the current expectation value to second order in the electric field ($E$) and disorder potential.
  • Decomposed the total current into three main physical components:
    1. Band current ($j^b$): Intrinsic contribution from unperturbed states.
    2. Side-jump current ($j^{sj}$): Disorder-induced correction analogous to side-jump velocity.
    3. Anomalous current ($j^a$): Field-induced correction analogous to anomalous velocity.
• Scattering Mechanisms:
  • The distribution function was decomposed into contributions from symmetric scattering, coordinate shift, anomalous scattering amplitude, and skew scattering (conventional and Gaussian).
  • The authors identified specific combinations of these terms that yield $T$-odd second-order responses.
• Model Calculation:
  • Employed a tilted four-band Dirac model that breaks both Parity ($P$) and Time-reversal ($T$) symmetries individually but preserves $PT$ symmetry.
  • Considered short-range random scalar impurities and phonon scattering.
  • Calculated analytical expressions for various conductivity terms in the limit of small tilt ($w \ll v$).

3. Key Contributions

• Unified Theory of $T$-odd Nonlinear Transport: The paper establishes a general theory for second-order $T$-odd angular-momentum Hall effects, explicitly separating spin and orbital contributions.
• Identification of New Disorder Mechanisms: Beyond the known Berry curvature dipole (ABD), the authors identified four distinct disorder-induced mechanisms contributing to the $T$-odd nonlinear current:
  1. Side-jump Angular-Momentum Current (SJC): Arising from the interplay of side-jump velocity and higher-order Drude shifts.
  2. Coordinate Shift (CS): Arising from field-induced coordinate shifts in the distribution function.
  3. Anomalous Scattering Amplitude (ASA): Arising from the antisymmetric part of the scattering amplitude.
  4. Skew Scattering (SK): Including both conventional and Gaussian skew scattering.
• General Scaling Law: The authors derived a general scaling relation for the second-order angular-momentum conductivity ($\chi$) with respect to longitudinal resistivity ($\rho$) and specific disorder resistivities ($\rho_i$):
  $$\chi = \frac{C}{\rho} + \sum_{i \in S} \frac{A_i}{\rho_i \rho^3} + \sum_{i} \frac{C_i}{\rho_i \rho^2} + \sum_{i,j} \frac{C_{ij}}{\rho_i \rho_j \rho^3}$$
  This allows experimentalists to distinguish between intrinsic ($1/\rho$), side-jump/coordinate shift ($1/\rho^2$), and skew scattering ($1/\rho^3$) mechanisms.
• Orbital Dominance: The theory demonstrates that the orbital component of the nonlinear Hall effect can be comparable to, and in certain regimes (small band gaps), significantly larger than the spin component.

4. Key Results

• Model Calculations (Tilted Dirac Model):
  • Analytical expressions were derived for the conductivity components ($\chi_{xyy}$) for both spin ($S_z$) and orbital ($L_z$) channels.
  • Scaling with Band Gap ($\Delta$): The ratio of orbital to spin conductivity scales as $\Delta^{-1}$. Specifically, for the Berry curvature dipole contribution:
    $$\frac{\chi^{L_z, ABD}_{xyy}}{\chi^{S_z, ABD}_{xyy}} \approx \frac{5ev^2}{3g_L \mu_B \hbar} \frac{1}{\Delta}$$
  • This implies that in small-gap systems, the nonlinear orbital Hall effect (NOHE) dominates the nonlinear spin Hall effect (NSHE).
• Disorder Dependence:
  • All identified mechanisms (ABD, SJC, CS, ASA, SK) contribute appreciably within the relevant energy window.
  • Reducing the band gap enhances both NSHE and NOHE, but the orbital contribution becomes increasingly dominant near the gap.
• Symmetry Considerations:
  • The $T$-odd nonlinear spin Hall effect can exist without spin-orbit coupling (SOC) in collinear magnetic structures (staggered exchange fields).
  • In contrast, the $T$-odd nonlinear orbital Hall effect requires SOC.
  • The authors suggest $PT$-symmetric antiferromagnets (e.g., CuMnAs, MnBi$_2$Te$_4$) as ideal candidates to probe these effects, as $T$-even contributions are forbidden in these systems.

5. Significance

• Fundamental Physics: The work fills a critical gap in the understanding of nonlinear transport by unifying spin and orbital degrees of freedom and rigorously incorporating disorder effects. It clarifies the microscopic origins of $T$-odd responses.
• Experimental Guidance: The derived scaling law provides a practical diagnostic tool for experimentalists to disentangle intrinsic and extrinsic contributions in nonlinear transport measurements, similar to how scaling laws are used in linear anomalous Hall effect studies.
• Material Design: The finding that orbital angular momentum can dominate nonlinear transport in small-gap systems opens new avenues for orbitaltronics. It suggests that materials with small band gaps and strong SOC are promising platforms for generating and controlling large orbital currents, potentially surpassing spin-based mechanisms in efficiency.
• Future Applications: The identification of specific material classes (PT-symmetric antiferromagnets) offers a concrete target for experimental verification of these theoretical predictions, potentially leading to new spintronic and orbitaltronic devices.