Fine structure of the cyclotron resonance in heterobilayers of proximitized graphene and transition metal dichalcogenides

Technical Summary: Fine Structure of Cyclotron Resonance in Proximitized Graphene/TMD Heterobilayers

Problem and Motivation
While pristine monolayer graphene possesses negligible intrinsic spin-orbit coupling (SOC), proximity to transition metal dichalcogenides (TMDs) or magnetic layers induces significant SOC, enabling new spintronic functionalities. Although spin relaxation, spin-charge interconversion, and spin Hall effects in such heterostructures have been extensively studied, the resonant high-frequency properties of these systems under quantizing magnetic fields remain underexplored. Specifically, the behavior of cyclotron resonance (CR) and combined cyclotron resonance (CCR)—involving SOC-induced spin-flip transitions between Landau levels (LLs)—has not been fully elucidated for graphene-based structures, particularly in contrast to the well-studied parabolic spectrum cases. This work addresses the theoretical analysis of CR absorption in monolayer graphene proximitized by TMDs, focusing on the emergence of fine structures and spin-flip absorption features driven by proximity-induced interactions.

Methodology
The authors employ a low-energy effective Dirac model augmented by proximity-induced interaction terms, consistent with first-principles studies of graphene/TMD heterostructures. The unperturbed Hamiltonian describes Dirac electrons around the $K$ and $K'$ valleys. The proximity effects are modeled via three additional terms:

1. Valley-Zeeman coupling ($H_{vz}$): Induced by the large intrinsic SOC of TMDs, causing valley-dependent spin splitting.
2. Staggered potential ($H_{st}$): Arising from sublattice asymmetry, opening an energy gap.
3. Rashba coupling ($H_R$): Due to substrate-induced inversion symmetry breaking, introducing a spin-pseudospin coupling (SPC) where the spin operator couples to the pseudospin (sublattice) rather than momentum.

The analysis assumes moderate magnetic fields (0.5–2 T) where cyclotron energies exceed proximity-induced terms. The authors start with the exact Landau level solutions of unmodified graphene and treat the proximity terms, particularly $H_R$, as perturbations. They perform exact numerical diagonalization of a truncated system (30 Landau levels) to account for the mixing of states with opposite spins and orbital indices. The absorption coefficient $\alpha(\hbar\omega)$ is calculated using the Kubo-Greenwood formula, incorporating the velocity operator matrix elements derived within the electric-dipole approximation (EDA).

Key Contributions and Results
The study reveals several distinct modifications to the cyclotron absorption spectrum caused by proximity-induced SOC:

1. Fine Double-Peak Structure: The main interband CR transitions, which are typically single lines in pristine graphene, develop a fine double-peak structure. This arises because the Rashba SOC ($H_R$) mixes Landau levels with opposite spin states and orbital indices differing by one ($n \to n \pm 1$). When combined with the valley-Zeeman splitting, this mixing results in spin-dependent energy shifts of the LLs. Consequently, transitions between LLs with the same orbital indices but opposite spins (e.g., $L^\uparrow_{-n} \to L^\uparrow_{n+1}$ and $L^\downarrow_{-n} \to L^\downarrow_{n+1}$) occur at slightly different resonant frequencies, creating a polarization-dependent double-peak fine structure.

2. Combined Cyclotron Resonance (CCR) and Spin-Flip Transitions: The authors identify the activation of CCR transitions, which involve spin-flip processes between LLs. Unlike standard CR where the orbital index changes by $\pm 1$, the selection rules for CCR in the presence of spin-pseudospin coupling allow orbital indices to change by $\Delta n \in \{0, \pm 2\}$.

   • Selection Rules: For $\sigma^+$ polarization, transitions include $n \to n+1$ (standard) and spin-flip transitions $n \uparrow \to n \downarrow$ or $n \downarrow \to n+2 \uparrow$. For $\sigma^-$, transitions include $n \to n-1$ and spin-flip transitions $n \downarrow \to n \uparrow$ or $n \uparrow \to n-2 \downarrow$.
   • Spectral Features: These spin-flip transitions appear as additional absorption lines or shoulders preceding the main CR interband series. Their energy positions differ from the main CR lines, leading to shape deformations in the absorption spectrum.

3. Polarization Dependence and MOKE: The absorption spectrum exhibits distinct polarization dependence ($\sigma^+$ vs. $\sigma^-$) due to the spin-dependent shifts and the specific selection rules of the CCR. This asymmetry leads to a finite magneto-optical Kerr effect (MOKE) around the peaks of interband CR absorption, a feature absent in pristine graphene where transitions for opposite polarizations are symmetric.

4. Isolation of CCR Lines: The paper identifies specific filling factors (e.g., $\nu = 4 \times l$) where the energy gap between intraband and interband CR transitions allows for the isolation of single CCR spin-flip absorption lines. This provides a potential experimental signature to distinguish spin-pseudospin coupling from linear-in-momentum Rashba interactions, as the latter would only induce spin-flip intraband transitions.

Significance
The paper claims that cyclotron resonance spectroscopy, particularly in the mid-infrared range, serves as a viable alternative method for elucidating proximity-induced SOC in graphene. By analyzing the fine double-peak structure and the presence of CCR spin-flip lines, one can experimentally verify the nature of spin interactions (specifically spin-pseudospin coupling) in graphene/TMD heterostructures. The theoretical framework provided explains the effect of proximity-induced spin interactions on Dirac electron cyclotron resonance and outlines specific spectral features (selection rules, polarization dependence, and MOKE) that can be targeted for experimental verification. The work highlights the versatility of tuning spin-dependent properties in proximitized graphene and offers a theoretical basis for interpreting high-frequency optical responses in these 2D heterostructures.