Lévy flight for electrons in graphene in the presence of regions with enhanced spin-orbit coupling

Technical Summary: Lévy Flight for Electrons in Graphene with Enhanced Spin-Orbit Coupling

Problem Statement
While pristine graphene possesses high electron mobility and gate tunability, its intrinsic spin-orbit coupling (SOC) is negligible, limiting its utility in spintronics. Although proximity effects with substrates (e.g., transition-metal dichalcogenides or topological insulators) can induce strong extrinsic SOC, the transport dynamics of electrons in graphene nanoribbons containing regions of enhanced SOC have not been fully characterized in the context of "Lévy glass" systems. Previous work established that graphene nanoribbons with circular electrostatic clusters (following a power-law radius distribution) exhibit a transition from superdiffusive to diffusive transport. This study investigates whether introducing tunable Rashba spin-orbit coupling in similar circular clusters induces analogous transport regimes and, crucially, how these regimes affect spin polarization and mesoscopic fluctuations.

Methodology
The authors model an electronic Lévy glass constructed from armchair (AGNR) and zigzag (ZGNR) graphene nanoribbons. The system consists of circular regions with enhanced Rashba SOC, randomly distributed across the ribbon. The radii of these circular clusters follow a heavy-tailed power-law distribution, $P(r) \propto r^{-(\beta+1)}$, with a cutoff to ensure physical realizability.

The transport properties are analyzed using the Landauer-Büttiker formalism within a tight-binding framework. The Hamiltonian includes nearest-neighbor hopping and a Bychkov-Rashba SOC term active only within the circular clusters. The scattering matrix is computed using the KWANT package to determine charge transmission coefficients and spin-dependent transmission matrices. From these, the authors calculate charge transmission ($T$) and spin polarization components ($P_x, P_y, P_z$).

To characterize the transport regime and mesoscopic fluctuations, the authors employ Multifractal Detrended Fluctuation Analysis (MF-DFA). In this analysis, the Fermi energy ($E$) serves as a "fictional time" variable. The scaling behavior of the transmission time series is examined to determine the generalized Hurst exponent, $h(q)$, and the multifractal singularity spectrum, $f(\alpha)$.

Key Results

1. Transport Regime Transition:
   The study confirms that spin-orbital clusters induce a transition from superdiffusive to diffusive charge transport as the Fermi energy increases (or equivalently, as the number of propagating modes $N$ increases).

   • Superdiffusive Regime ($N < 10$): At low energies, the scaling exponent $\gamma$ (where average transmission $\langle T \rangle \sim L^{-\gamma}$) is less than 1 (e.g., $\gamma \approx 0.30$ to $0.65$), indicating Lévy-like superdiffusive transport.
   • Diffusive Regime ($N > 10$): At higher energies, $\gamma$ approaches 1, signifying a transition to standard diffusive transport.
   • This transition is independent of the ribbon edge type (AGNR vs. ZGNR) and the specific strength of the Rashba interaction.

2. Spin Polarization Behavior:
   A distinct dichotomy is observed between charge transport and spin polarization:

   • Superdiffusive Regime: Finite and significant spin polarization is generated. The spin-orbital clusters induce an efficient separation of spin-up and spin-down states, resulting in high spin polarization efficiency.
   • Diffusive Regime: Spin polarization vanishes (approaches zero) regardless of device length or interaction strength. In this regime, spin-up and spin-down electrons are not segregated, leading to high magnetoresistivity and suppressed polarization.
   • Consequently, the device functions as a tunable spin filter, where the spin polarization can be controlled by adjusting the Fermi energy.

3. Multifractal Analysis:
   The application of MF-DFA reveals fundamental differences in the mesoscopic fluctuations of charge versus spin:

   • Charge Transmission: The time series is multifractal in the superdiffusive regime (broad $f(\alpha)$ spectrum) but tends toward monofractal behavior in the diffusive regime. Notably, a significant increase in multifractality is observed near the transition point ($N \approx 7-10$), which the authors interpret as a signature of a non-equilibrium phase transition associated with chiral symmetry breaking.
   • Spin Polarization: In contrast, the spin polarization time series remains multifractal in both the superdiffusive and diffusive regimes. This indicates that the mesoscopic fluctuations of spin polarization are robust against the transition to diffusive transport, unlike charge transport fluctuations.

Significance and Claims
The paper claims to demonstrate that electronic Lévy glasses built with spin-orbital clusters are viable candidates for spintronic applications. The primary significance lies in the ability to control both electronic transmission and spin polarization via the Fermi energy. Specifically, the device acts as an efficient spin filter only in the superdiffusive regime (low energy), while the spin polarization vanishes in the diffusive regime.

Furthermore, the work provides a theoretical framework linking the superdiffusive-to-diffusive transition to chiral symmetry breaking, evidenced by the multifractal analysis of charge transmission. The authors highlight a marked difference in the nature of mesoscopic fluctuations between charge and spin: while charge transport loses its multifractal character upon entering the diffusive regime, spin polarization retains multifractality. This suggests that spin polarization is more sensitive to density of states variations and correlations induced by the SOC clusters than pure charge transport.

The study extends previous findings on electrostatic Lévy glasses to the spintronic domain, confirming that spin-orbital clusters can independently induce Lévy-type transport and offering a mechanism to tune spin polarization in graphene-based nanostructures.