Valley Valves at Domain Walls in Symmetry-Broken Rhombohedral Graphene

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

Problem Statement
Recent experiments on rhombohedral multilayer graphene under moderate perpendicular displacement fields have revealed a symmetry-broken "quarter-metal" phase that is fully valley- and spin-polarized. This state is believed to condense into a chiral superconductor. Crucially, magnetic imaging and transport measurements suggest the existence of real-space domain walls separating regions of opposite valley polarization (e.g., $K$ vs. $K'$ valleys). The central problem addressed in this work is understanding the transport properties across these valley domain walls. Specifically, the authors investigate whether electrons can transmit ballistically between valley-contrasting domains and, if not, what mechanisms are required to mediate transport in both the metallic and superconducting regimes.

Methodology
The authors employ a combination of analytical theory and microscopic numerical simulations to model transport across domain walls in $N$-layer rhombohedral graphene.

1. Analytical Modeling:

   • They begin with a low-energy effective Hamiltonian for spin-polarized monolayer and $N$-layer rhombohedral graphene, incorporating valley-dependent sublattice asymmetry ($\varepsilon_K$ and $\varepsilon_{K'}$) to model the valley polarization.
   • They derive analytic solutions for scattering states at an abrupt domain wall, enforcing wavefunction continuity and current conservation.
   • They generalize these results to $N$-layer systems using a minimal two-band model that captures the $k^N$ dispersion, analyzing the behavior of propagating and evanescent modes.

2. Symmetry Classification:

   • The authors classify symmetry-allowed intervalley-mixing terms for a reconstructed $\sqrt{3} \times \sqrt{3}$ Kekulé lattice. They distinguish between intravalley and intervalley operators based on the number of layers $N$ (specifically $N \mod 3$) and the effective lattice symmetry (triangular vs. honeycomb).
   • They identify generic intervalley terms ($\tau_x\sigma_+$ and $\tau_y\sigma_+$) that are symmetry-allowed for all $N$ and act on the active $A(1)$ sublattice.

3. Numerical Simulations:

   • Using a tight-binding Hamiltonian with five hopping parameters ($\gamma_0$ to $\gamma_4$) and a layer-dependent displacement field, they model $N=4$ rhombohedral graphene.
   • They utilize the Green's function formalism (Caroli-Fisher-Lee formula) to calculate transmission functions ($T(E, k_y)$) for finite-width domain walls.
   • They simulate the introduction of intervalley coupling ($m$) within the domain wall region to test its effect on transmission.
   • For the superconducting phase, they construct a Bogoliubov-de Gennes (BdG) Hamiltonian assuming $p$-wave pairing ($p_+$ in the $K$ valley and $p_-$ in the $K'$ valley) and calculate the DC supercurrent across an SNS' Josephson junction.

Key Contributions and Results

• Opacity of Abrupt Domain Walls:
  The authors demonstrate analytically that an abrupt domain wall separating opposite valley polarizations is perfectly opaque to electron transmission in the metallic phase. Even for armchair-oriented walls where valleys map to similar crystal momenta (kinematically allowing mixing), the absence of intervalley coherence forces total reflection. The incoming wave is reflected into the same valley, while evanescent modes from the opposite valley carry current parallel to the wall but do not transmit across it. This result holds for both monolayer and $N$-layer systems.

• Valley Mixing as a Transport Valve:
  Numerical simulations confirm that transmission across a finite-width domain wall remains zero unless intervalley coupling is introduced. The authors show that specific symmetry-allowed terms (specifically $\tau_x\sigma_+$ and $\tau_y\sigma_+$) hybridize the valleys.

  • Transmission Dependence: Transmission increases monotonically with the strength of the intervalley coupling ($m$) for moderate wall widths. For small widths, the dependence is quadratic; for larger widths, an abrupt switching behavior is observed where transmission jumps to unity once a threshold coupling is reached.
  • Angle Independence: The transmission is largely insensitive to the angle $\theta$ of the intervalley interaction in $(\tau_x, \tau_y)$ space.
  • Filling Dependence: In the quarter-metal phase, intervalley coupling enables transmission. However, in regimes where both valleys are occupied, intervalley coupling can introduce scattering that reduces transmission, acting as a valve that controls transport based on the electronic phase.

• Supercurrent in Chiral Superconductors:
  In the superconducting phase, the authors model a Josephson junction connecting a $p_+$ superconductor ($K$ valley) to a $p_-$ superconductor ($K'$ valley) via a metallic domain wall.

  • They find that without intervalley mixing, the supercurrent is heavily suppressed (though not numerically zero due to finite numerical resolution, it is negligible).
  • The presence of intervalley mixing in the normal-state domain wall is crucial for sustaining an appreciable supercurrent. The supercurrent density scales with the transmission transparency of the normal state, consistent with the Ambegaokar-Baratoff relation.
  • The current-phase relation (CPR) oscillates in correspondence with the oscillation of Majorana branches traversing the BdG gap, indicating that Andreev bound states dominate the supercurrent.

Significance and Claims
The paper claims to elucidate the nature of domain walls in symmetry-broken rhombohedral graphene, emphasizing that valley domain walls act as "valve" barriers. The primary significance lies in the finding that intervalley hybridization is the critical mechanism governing transport across these walls.

• Valve Mechanism: The authors posit that the domain wall itself is an impenetrable barrier in the absence of intervalley mixing. Transport is only possible if the wall contains defects, disorder, or specific interactions that mix the valleys.
• Experimental Relevance: The results provide a theoretical framework for interpreting recent experimental observations of hysteretic transport and superconductivity in these systems. The authors suggest that the observed transport signatures likely arise from intervalley mixing mediated by impurities or specific structural features at the domain walls.
• Chiral Superconductivity: The work highlights that the transport of supercurrent between opposite-chirality superconducting regions is fundamentally dependent on the normal-state intervalley coherence, linking the properties of the normal and superconducting phases through the domain wall physics.

The authors conclude that resolving the microscopic origin of this intervalley mixing remains an open problem essential for fully understanding the physics of these novel correlated states.