Valley Valves at Domain Walls in Symmetry-Broken Rhombohedral Graphene

This paper demonstrates that in symmetry-broken rhombohedral graphene, valley domain walls act as impenetrable barriers in the metallic regime and require intervalley mixing to support supercurrents in the superconducting phase, highlighting the critical role of intervalley hybridization in governing transport across these boundaries.

The Gist

Imagine a piece of graphene (a material made of a single layer of carbon atoms) that has been stacked up like a sandwich. When you apply a specific electric field to this stack, something magical happens: the electrons inside get organized into two distinct "teams" based on a property called "valley." Think of these valleys like two different sports teams, Team K and Team K', playing on the same field.

In this special state, the electrons are so polarized that they only belong to one team or the other, never both. The paper explores what happens when these two teams meet at a boundary line, known as a domain wall.

Here is the story of what the researchers found, explained simply:

1. The "No-Entry" Sign

The researchers discovered that if you try to send an electron from Team K's side of the field to Team K' side, it hits a brick wall. It's like trying to drive a car from a country where you drive on the right directly into a country where you drive on the left, but without a bridge or a tunnel.

In the "metallic" state (where electricity flows like water), the domain wall is impenetrable. An electron coming from the K side hits the wall and bounces straight back, staying on the K side. It cannot cross over to the K' side. The wall acts as a perfect mirror for these electrons.

2. The Secret Door: Intervalley Mixing

So, how do the electrons ever get across? The paper explains that there is a "secret door" that only opens if the two teams start talking to each other. This interaction is called intervalley mixing.

Imagine that normally, Team K and Team K' speak different languages and ignore each other. But if you introduce a translator (which the researchers model as a specific type of atomic disturbance or "impurity" at the wall), the teams can understand each other. Once this "translator" is present, the electrons can finally cross the wall.

The researchers used computer simulations to show that without this mixing, the wall is 100% opaque. With the mixing, the wall becomes transparent, and the electrons can flow through.

3. The Superconducting "Super-Current"

The paper also looked at what happens when the material becomes a superconductor (a material that conducts electricity with zero resistance). In this state, electrons pair up to form "Cooper pairs" and flow like a super-fluid.

The researchers found that even in this super-state, the wall is still a barrier unless the "translator" (intervalley mixing) is present.

• Without the translator: The super-current is almost zero. The wall blocks the super-flow.
• With the translator: The super-current flows strongly across the wall.

It's like a river trying to flow through a dam. If the dam is solid, the water stops. But if you install a specific type of valve (the intervalley mixing), the water rushes through.

4. Why This Matters

The main takeaway is that in these special graphene stacks, the "valley" property acts like a gatekeeper.

• The Gate: The domain wall.
• The Gatekeeper: The lack of communication between the two valleys.
• The Key: Intervalley mixing.

The paper concludes that to understand how electricity (both normal and super) moves through these materials, you have to understand this "valley valve." If you want the current to flow, you need to engineer the material so that the two valley teams can interact at the boundary. Without that interaction, the wall remains a perfect blockade.

In short: The paper maps out a new kind of traffic jam in graphene where electrons get stuck at a boundary line unless a specific "mixing" mechanism is introduced to let them pass through. This mixing is the critical key to unlocking transport in these advanced materials.

Technical Summary

Technical Summary: Valley Valves at Domain Walls in Symmetry-Broken Rhombohedral Graphene

Problem Statement
Rhombohedral multilayer graphene, when subjected to a moderate perpendicular displacement field, hosts a symmetry-broken metallic phase that is fully valley-and-spin polarized (a "quarter metal"). This state can condense into a chiral superconductor. Recent experimental observations suggest the existence of real-space domain walls separating regions of opposite valley polarization in both the metallic and superconducting phases. A critical open question is how electron transport occurs across these domain walls. While naive expectations might suggest that aligning momenta of valley-contrasting states allows transmission, the authors investigate whether these walls act as barriers and what mechanisms are required to mediate transport between valley-polarized domains.

Methodology
The authors employ a combination of analytical continuum modeling and microscopic numerical simulations using tight-binding Hamiltonians and Green's function formalisms.

1. Analytical Continuum Models: The authors first derive analytic solutions for an abrupt domain wall separating two regions of opposite valley polarization. They utilize low-energy effective Hamiltonians for both monolayer and $N$-layer rhombohedral graphene, incorporating valley-dependent mass gaps ($\varepsilon_K$ and $\varepsilon_{K'}$) that switch sign across the wall. They enforce boundary conditions via wavefunction continuity and current conservation to determine reflection and transmission amplitudes.
2. Symmetry Classification: To understand how transport might be enabled, the authors classify symmetry-allowed intervalley mixing terms. They consider Kekulé-type $\sqrt{3} \times \sqrt{3}$ lattice reconstructions that fold the $K$ and $K'$ valleys to the same point in the Brillouin zone ($\bar{\Gamma}$). They identify specific operators (e.g., $\tau_x\sigma_+$ and $\tau_y\sigma_+$) that are allowed by $C_3$ rotation symmetry and can hybridize the valleys.
3. Microscopic Numerical Simulations: Using a standard tight-binding model for $N$-layer rhombohedral graphene (including interlayer hopping parameters $\gamma_0$ through $\gamma_4$ and a displacement field $\Delta$), the authors simulate scattering across finite-width domain walls. They employ the Caroli-Fisher-Lee formula to calculate transmission functions $T(E, k_y)$ for normal and oblique incidence.
4. Superconducting Transport: For the superconducting phase, they model a Josephson junction (SNS' structure) connecting a $p_+$ superconductor in the $K$ valley to a $p_-$ superconductor in the $K'$ valley. They compute the DC supercurrent using the phase-dependent free energy and the BdG (Bogoliubov-de Gennes) Green's function formalism.

Key Contributions and Results

• Opacity of Abrupt Domain Walls: The authors analytically demonstrate that an abrupt domain wall separating opposite valley-polarized quarter metals is perfectly opaque to electron transmission. In the metallic regime, an incoming wave from one valley (e.g., $K$) is completely reflected back into the same valley. While evanescent modes exist in the opposite valley, they do not carry current across the wall in the absence of intervalley coupling. This result holds for both monolayer and $N$-layer systems, regardless of the wall's orientation (zigzag or armchair).
• Role of Intervalley Hybridization: The study establishes that transport across such domain walls is strictly forbidden without intervalley mixing. The authors identify specific symmetry-allowed terms (specifically $\tau_x\sigma_+$ and $\tau_y\sigma_+$ acting on the active sublattice) that can hybridize the valleys. Numerical simulations confirm that introducing these coupling terms within the domain wall enables robust electron transmission. The transmission increases with the strength of the intervalley coupling $m$, reaching unity at resonant conditions for specific wall widths.
• Superconducting Transport: In the chiral superconducting phase, the authors find that intervalley mixing is equally crucial. They model a Josephson junction connecting opposite-chirality superconductors ($p_+$ and $p_-$). Without intervalley coupling, the supercurrent is heavily suppressed (though not numerically zero due to finite broadening). As the intervalley coupling strength increases, the supercurrent density is appreciably amplified, consistent with the Ambegaokar-Baratoff relation. The supercurrent is found to be dominated by Andreev bound states and Majorana branches traversing the junction.
• Valve Mechanism: The results illustrate that intervalley interactions act as a "valve" controlling transport. In the absence of these interactions, domain walls are impenetrable barriers; with them, transport is restored.

Significance and Claims
The paper elucidates the nature of domain walls in experimentally relevant rhombohedral multilayer graphene systems. The authors claim that their work explains recent experimental observations of transport signatures between valley-polarized domains, suggesting that such transport must be mediated by some form of intervalley mixing (potentially via impurities or disorder at the domain walls).

A central claim is that valley domain walls are fundamentally transport blockades in the absence of intervalley coherence. This "valve" behavior is critical for understanding the hysteretic transport and magnetic training effects observed in these systems. The work emphasizes that the microscopic origin of intervalley hybridization is the key factor governing the transport properties of both the normal metallic and chiral superconducting phases in these symmetry-broken systems. The authors note that while they have modeled the effect phenomenologically, resolving the specific microscopic origin of the mixing in experiments remains an open problem.