Persistent current and orbital magnetization along a valley-contrasting junction in bilayer graphene in a magnetic field

Technical Summary: Persistent Current and Orbital Magnetization along a Valley-Contrasting Junction in Bilayer Graphene

Problem Statement
Bilayer graphene in a magnetic field hosts an octet of pseudo-zero-mode (PZM) electron levels (comprising orbitals $n=0, 1$, two valleys, and two spins) that are nearly degenerate. While a uniform interlayer bias opens a tunable valley band gap, a specific configuration involving a line junction where the bias flips sign creates a unique interface. This interface traps PZM electrons within the insulating bulk band gap, forming localized "kink states." While the existence and transport properties of these kink states have been established theoretically and experimentally, their equilibrium electromagnetic response—specifically the nature of persistent currents and orbital magnetization—remains less explored. This paper investigates the detailed spectra and electromagnetic features of these kink states, focusing on the interplay between drift currents and circulating currents in the quantum Hall regime.

Methodology
The author employs an algebraic approach to diagonalize the PZM levels under a spatially varying interlayer bias $\mu(y)$. The analysis utilizes a four-component spinor formalism for bilayer graphene, incorporating the effective Hamiltonian with leading couplings ($\gamma_0, \gamma_1$) and a position-dependent bias.

• Spectral Analysis: The bias profile is modeled as $\mu(y) = -\mu \tanh(y/w_{jc})$, representing a sharp reversal across a junction. The Hamiltonian is diagonalized in the orbital basis to determine the energy spectra of the mixed kink states.
• Electromagnetic Response: A $(W_\infty)$ gauge theory framework is used to derive real-space distributions of current and orbital magnetization. The method involves a unitary gauge transformation to separate the Hamiltonian into diagonal energy spectra and off-diagonal terms.
• Current Decomposition: The total current is decomposed into two components: a drift current ($j^{(d)}$) driven by the bias gradient (electric field), and a circulating current ($j^{(c)}$) arising from cyclotron motion. The analysis calculates these currents for both individual valleys ($K, K'$) and the full system in equilibrium.

Key Contributions and Results

1. Orbital Mixing and Spectra: The study confirms that across the valley-contrasting junction, the PZM levels undergo orbital mixing (e.g., $(1^-, 0^-) \to (0^+, 1^+)$ in valley $K$) to avoid level crossing, characterized by a non-Abelian Berry phase of $\pi$. This mixing creates four active channels (per spin) of electron modes trapped within the bulk band gap.
2. Unidirectional Flow of Currents: A central finding is that in the kink states, both the drift current and the circulating current flow in essentially the same direction on each side of the junction. This contrasts with the behavior in higher Landau levels or standard edge states, where drift and circulating currents often flow in opposite directions (leading to de Haas-van Alphen oscillations).
3. Magnetic Character of PZM Electrons: The circulating current exhibits an unusual magnetic character. Specifically, the $n=1$ PZM electrons in the positive-energy branch display a paramagnetic response (positive magnetic moment), whereas standard positive-energy electrons are typically diamagnetic. This paramagnetism arises because the energy of these pseudo-zero modes decreases as the magnetic field increases.
4. Orbital Magnetization Channels: In equilibrium, the spatial circulation of the total current ($j^{(d)} + j^{(c)}$) creates a quasi-one-dimensional channel of orbital magnetization localized along the junction. The drift component dominates the magnetization, which is paramagnetic. The circulating component adds a small peak profile near the edges of the filled domain.
5. Valley Contrast and Control: The charge and current densities of the $K$ and $K'$ valleys overlap in the central region of the junction but remain distinct in the outer regions. The valley contrast becomes more pronounced with increasing electron filling and magnetic field. The direction of the valley currents and the associated orbital magnetization can be controlled by reversing the polarity of the gating (interlayer bias).

Significance and Claims
The paper claims that the kink states in gated bilayer graphene provide a robust platform for generating persistent currents and localized orbital magnetization without the need for injected currents. The primary significance lies in the demonstration that a network of such valley-contrasting junctions can form a network of narrow, electrically controllable orbital magnetization paths.

The author posits that the ability to electrically control these orbital magnetization channels and the associated valley currents via a network of gated junctions holds potential utility for "valley electronics" (or valleytronics). The work suggests that signals in such a network could be carried by the equilibrium orbital magnetization itself, detectable via local magnetization measurements at the network's ends. The paper concludes that while the drift current drives the valley Hall effect, the unique magnetic response of the PZM electrons ensures that the circulating current reinforces rather than cancels the drift current, stabilizing the magnetization channel against disorder as long as the bias profile is maintained.