Magnetotransport across Weyl semimetal grain boundaries

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This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A Traffic Jam on a Magic Highway

Imagine a Weyl Semimetal as a magical city where electrons (the cars) don't just drive in straight lines; they follow special, protected paths called Fermi arcs. These paths are like one-way highways that connect different parts of the city. Because of the laws of quantum physics, these highways are very hard to block.

Now, imagine you have two of these magical cities, and you build a bridge between them. This bridge is the interface. In a perfect world (a "clean" interface), the cars can zip across this bridge effortlessly, but only if they are going in the right direction.

The scientists in this paper wanted to answer a big question: What happens if the bridge is messy? What if there are potholes, debris, or random obstacles (which they call "disorder") on the bridge? Does the traffic stop? Does the magic disappear?

The Key Discovery: Two Speed Limits

The researchers found that the traffic flow (conductance) depends entirely on how strong the magnetic field is. Think of the magnetic field as a strong wind blowing across the bridge.

1. The "Super-Wind" Scenario (High Magnetic Field)

Imagine a hurricane-force wind blowing across the bridge.

What happens: The wind is so strong that it pushes the cars so fast that they don't have time to hit the potholes or get distracted by the debris. They zoom straight across the bridge, ignoring the mess.
The Result: The traffic flow is perfect. It behaves exactly as if the bridge were brand new and clean. The number of cars passing through depends only on how many "magic lanes" (Fermi arcs) exist.
The Analogy: It's like running through a crowded room while someone is blowing a giant fan at your back. You move so fast that you don't bump into anyone.

2. The "Gentle Breeze" Scenario (Low Magnetic Field)

Now, imagine the wind is very light, just a gentle breeze.

What happens: The cars are moving slowly. They have plenty of time to hit the potholes, bounce off the debris, and get confused. They might even switch lanes or get stuck in a traffic jam.
The Result: The traffic flow drops significantly. It doesn't matter how many "magic lanes" you have; the flow is now limited by how messy the bridge is.
The Surprise: The researchers found a specific rule for this low-wind traffic. The flow becomes a simple fraction based on the number of entry and exit points on both sides of the bridge. It's like a math puzzle where the answer is always "half" (or a specific ratio) of the maximum possible flow, regardless of the specific details of the mess.

The "Tipping Point" ( $B_{arc}$ )

The paper introduces a special number called $B_{arc}$ . Think of this as the Tipping Point Wind Speed.

Above $B_{arc}$ : The wind is strong enough to overcome the mess. The system acts "clean."
Below $B_{arc}$ : The wind is too weak. The mess takes over, and the traffic slows down to that specific "fractional" limit.

Why Does This Matter? (The Grainy Mystery)

Recently, other scientists looked at grained Weyl Semimetals. Imagine a material made of many tiny, randomly oriented crystals (like a pile of sand) rather than one big perfect crystal. When they tested these materials, they saw something strange: the electricity flowed in a straight line (linearly) as they increased the magnetic field, and this behavior was very robust (it didn't break even when the material was messy).

This paper explains why.
The "grains" in the material act like millions of tiny bridges (interfaces) between the crystals.

Because the magnetic field is usually strong enough to be above the "Tipping Point" ( $B_{arc}$ ) for these tiny bridges, the messiness of the grain boundaries doesn't matter.
The electrons just zoom across the grain boundaries, preserving the special "chiral" flow.
This explains why the material works so well even when it's not a perfect crystal.

The "Smooth vs. Rough" Mess

The paper also looked at what kind of mess was on the bridge.

Rough Mess (Random Potholes): If the debris is scattered randomly, it causes chaos.
Smooth Mess (Long, rolling hills): If the debris is smooth and connected (correlated disorder), it's actually less annoying! The cars can glide over the hills without getting stuck.
The Finding: The more "smooth" and connected the mess is, the lower the "Tipping Point" wind speed becomes. This means the system stays "perfect" even in weaker winds if the mess is smooth enough.

Summary

The Magic: Electrons in Weyl Semimetals travel on special "Fermi arc" highways.
The Problem: Real materials have defects (disorder) that usually stop traffic.
The Solution: A strong magnetic field acts like a super-wind that pushes electrons so fast they ignore the defects.
The Twist: If the wind is too weak, the traffic slows down to a predictable, simple fraction of the maximum speed.
The Real-World Win: This explains why "grainy" (imperfect) Weyl Semimetals still show amazing electrical properties. The magnetic field is strong enough to keep the traffic flowing smoothly across the messy grain boundaries.

In short: Strong magnetic fields turn a messy, broken bridge into a super-highway.

1. Problem Statement

Weyl semimetals (WSMs) exhibit anomalous transport phenomena, such as the chiral magnetic effect (CME), driven by topologically protected Weyl nodes and surface Fermi arcs. While the CME is theoretically predicted to manifest as a universal, linear tunnel magnetoconductance across clean interfaces between WSMs, experimental observations in grained WSMs (polycrystalline materials with randomly oriented crystallites) show a robust negative linear magnetoresistance.

The central problem addressed in this work is the robustness of this interface-mediated magnetotransport in the presence of disorder. Specifically, the authors investigate:

How interface disorder (scattering) affects the transmission of chiral Landau levels via Fermi arcs.
Whether the universal linear magnetoconductance ( $G \propto B$ ) persists in disordered systems.
How the transport behavior changes across different magnetic field regimes relative to a characteristic disorder scale.

2. Methodology

The authors employ a hybrid theoretical approach combining analytical formalisms with numerical simulations:

Analytical Framework:
- Landauer-Büttiker Formalism: Used to define the conductance in terms of transmission probabilities of chiral Landau levels.
- Semiclassical Boltzmann Transport Equation (BTE): Applied to the Fermi arcs at the interface to model the dynamics of non-equilibrium occupation functions under the influence of the Lorentz force and disorder scattering.
- Scattering Models: The disorder is modeled via a random onsite potential with specific autocorrelation functions to distinguish between uncorrelated (point-like) and correlated (spatially smooth) disorder.
Numerical Simulations:
- Tight-Binding Model: A minimal lattice model representing two WSMs with shifted Weyl nodes across an interface ( $x=0$ ).
- Kwant Package: Used for full-quantum transport simulations to calculate conductance across disordered interfaces.
- Validation: The analytical predictions derived from the hybrid Landauer-Boltzmann approach are benchmarked against the full-quantum numerical results.

3. Key Contributions and Theoretical Findings

A. The Characteristic Field Scale ( $B_{arc}$ )

The authors identify a critical magnetic field strength, $B_{arc}$ , which marks the crossover between two distinct transport regimes. $B_{arc}$ is defined by the condition where the dwell time ( $\tau_{dwell}$ ) of a particle on a Fermi arc (driven by the Lorentz force) equals the mean inter-arc scattering time ( $\tau_{arc}$ ):
$\tau_{dwell} = \tau_{arc}$

$\tau_{dwell} \propto 1/B$ (decreases with increasing field).
$\tau_{arc}$ is determined by the disorder strength and correlation length.

B. Two Distinct Transport Regimes

The magnetoconductance $G(B)$ exhibits linear behavior in both limits but with different slopes:

High-Field Regime ( $B \gg B_{arc}$ ):
- The dwell time is much shorter than the scattering time ( $\tau_{dwell} \ll \tau_{arc}$ ).
- Particles traverse the Fermi arc before scattering significantly.
- Result: The system recovers the clean-interface limit. The magnetoconductance is universal and determined solely by the number of homochiral (chirality-preserving) Fermi arc pairs ( $N_{ho}$ ):
  $G(B) \approx N_{ho} G_0(B)$
  where $G_0(B) = \frac{e^3}{h^2} |A \cdot B|$ .
Low-Field Regime ( $B \ll B_{arc}$ ):
- The dwell time is much longer than the scattering time ( $\tau_{dwell} \gg \tau_{arc}$ ).
- Particles undergo multiple scattering events (both intra-arc and inter-arc) before exiting the interface.
- Result: The conductance becomes a universal fraction of the clean limit, determined by the total number of Weyl node pairs in the left ( $N_L$ ) and right ( $N_R$ ) WSMs, rather than the specific arc connectivity:
  $G(B) = \frac{N_L N_R}{N_L + N_R} G_0(B)$
- For a minimal model with one pair of nodes on each side ( $N_L=N_R=1$ ), the slope is exactly half of the high-field slope ( $G(B) = \frac{1}{2} G_0(B)$ ).

C. Effect of Disorder Correlation

The authors analyze spatially correlated disorder potentials. They find that the characteristic field $B_{arc}$ decreases exponentially with increasing disorder correlation length ( $\xi$ ).

Smooth disorder suppresses inter-arc scattering (scattering between counter-propagating channels).
As $\xi$ increases, the system effectively behaves as if it is cleaner, pushing the crossover to lower fields.

4. Numerical Results and Validation

Agreement: The numerical simulations using the Kwant package on a tight-binding model show excellent quantitative agreement with the analytical Landauer-Boltzmann predictions, even for relatively strong disorder ( $W=0.2$ ).
Transmission Probability: Simulations confirm that the transmission probability $T(B)$ transitions from $0.5$ (low field) to $1.0$ (high field) as the magnetic field increases.
Magnetic Breakdown: In the case of weakly coupled interfaces (where arcs are very close), quantum tunneling (magnetic breakdown) suppresses transmission. The authors show that disorder has a negligible effect on this suppression in the high-field limit.
Non-Chiral Contributions: The study confirms that contributions from non-chiral Landau levels (disorder-activated bulk modes) are negligible for well-separated Weyl nodes, justifying the focus on Fermi arcs.

5. Significance and Implications

Explanation of Experimental Data: The results provide a theoretical explanation for the robust negative linear magnetoresistance observed in grained WSMs (e.g., Pt $_3$ Sn). The observed factor-of-2 difference in magnetoresistance slopes between low and high fields matches the theoretical prediction of the transition from the $N_L N_R / (N_L + N_R)$ regime to the $N_{ho}$ regime.
Robustness of Topology: The work demonstrates that the topological nature of Fermi-arc mediated transport is remarkably robust against interface disorder, provided the magnetic field is sufficiently high.
Material Design: The findings suggest that the magnetoconductance behavior can be tuned by engineering the interface (controlling $N_{ho}$ , $N_L$ , $N_R$ ) and the disorder profile (correlation length), offering a pathway for designing topological electronic devices.
Methodological Advance: The development of a hybrid Landauer-Boltzmann theory offers an efficient tool for analyzing topological transport in disordered systems, bridging the gap between purely quantum simulations and semiclassical approximations.

In conclusion, the paper establishes that the linear magnetoresistance in Weyl semimetals is a robust topological feature that survives interface disorder, characterized by a distinct crossover in slope determined by the interplay between the Lorentz force and inter-arc scattering rates.