Symmetry-Based Design Rules for Second-Harmonic Generation in Stacked and Twisted MoS2 Bilayers

Technical Summary: Symmetry-Based Design Rules for Second-Harmonic Generation in Stacked and Twisted MoS2 Bilayers

Problem Statement
The nonlinear optical response of two-dimensional transition metal dichalcogenides (TMDs), particularly MoS$_2$, is highly sensitive to stacking configurations and interlayer rotation. While first-principles calculations can model idealized systems, they become computationally prohibitive for realistic low-symmetry configurations and large-scale moiré superlattices. There is a need for a rigorous, symmetry-based framework that can map structural configurations to their second-order susceptibility tensors ($\chi^{(2)}$) without relying on brute-force calculations for every specific twist angle or stacking variant.

Methodology
The authors establish a comprehensive framework combining group-theoretical analysis with targeted first-principles validations:

1. Symmetry Analysis: Using Neumann's principle, the authors derive the allowed and forbidden components of the $\chi^{(2)}$ tensor for four distinct point groups relevant to MoS$_2$ bilayers: $D_{3h}$ (monolayer and AA-stacked), $D_{3d}$ (AB(2H)-stacked), $C_{3v}$ (AB(3R)-stacked), and $C_3$ (twisted bilayers). They analyze how specific symmetry elements (inversion center $I$, horizontal mirror $\sigma_h$, vertical mirrors $\sigma_v$) constrain the tensor structure.
2. First-Principles Calculations: The theoretical predictions are benchmarked against density functional theory (DFT) calculations using the VASP code with the PBE functional and DFT-D3 corrections for van der Waals forces. The nonlinear optical response is computed using maximally localized Wannier functions (via Wannier90 and postw90) within the independent-particle approximation. The calculations include spin-orbit coupling and decompose the response into interband and intraband contributions.
3. Angular Response Modeling: The authors derive analytical expressions for the co-polarized second-harmonic generation (SHG) intensity $I_{\parallel}(\theta)$ as a function of the incident polarization angle $\theta$ for each symmetry class, specifically investigating the emergence of phase shifts in twisted configurations.

Key Contributions

• Symmetry-to-Response Mapping: The paper provides a definitive mapping of how stacking configurations (AA, AB(2H), AB(3R), and twisted) dictate the $\chi^{(2)}$ tensor structure across the $D_{3h}$, $D_{3d}$, $C_{3v}$, and $C_3$ point groups.
• Design Rules for SHG Suppression and Activation:
  • Inversion Symmetry ($D_{3d}$): The presence of an inversion center in AB(2H) stacking strictly enforces $\chi^{(2)} = 0$, leading to complete SHG suppression via destructive interlayer interference.
  • Horizontal Mirror Breaking ($C_{3v}$ and $C_3$): Breaking the $\sigma_h$ symmetry in AB(3R) and twisted bilayers activates out-of-plane tensor components ($\chi_{zzz}^{(2)}, \chi_{zxx}^{(2)}, \chi_{xxz}^{(2)}$), which are forbidden in $D_{3h}$ systems.
  • Vertical Mirror Breaking ($C_3$): The removal of vertical mirror planes in twisted bilayers activates the in-plane component $\chi_{xxx}^{(2)}$, which is strictly zero in higher-symmetry configurations.
• Geometric Locking of SHG Lobes: The authors demonstrate that the activation of $\chi_{xxx}^{(2)}$ in twisted bilayers induces a rigid azimuthal rotation of the SHG polar lobes. Crucially, they find that this rotation angle $\phi_0$ is frequency-independent across the optical spectrum and is geometrically locked to the structural twist angle $\theta_{tw}$ via the relation $\theta_{tw} = 2\phi_0$.

Results

• Tensor Validation: First-principles calculations confirm the group-theoretical predictions. The AB(2H) bilayer shows vanishing SHG signals. The AB(3R) bilayer recovers in-plane response similar to the monolayer but gains significant out-of-plane components. The twisted bilayer ($C_3$) exhibits simultaneous non-zero $\chi_{yyy}^{(2)}$ and $\chi_{xxx}^{(2)}$.
• Spectral Features: The SHG spectra are dominated by two-photon resonances corresponding to the A and B excitonic transitions (spin-orbit split valence bands) and a higher-energy C peak associated with band-nesting transitions.
• Polar Pattern Rotation: For a twisted bilayer with $\theta_{tw} = 21.8^\circ$, the SHG polar pattern rotates by $\phi_0 \approx 10.9^\circ$. This shift remains constant across the entire computed frequency range, confirming that the phase relationship between $\chi_{xxx}^{(2)}$ and $\chi_{yyy}^{(2)}$ is dictated by the structural geometry rather than dynamic spectral variations.
• Chiral Components: While the chiral out-of-plane component $\chi_{xyz}^{(2)}$ is symmetry-allowed in $C_3$, its magnitude is found to be exceptionally small due to the weak interlayer coupling and local preservation of mirror symmetry within individual monolayers.

Significance
The paper establishes a direct, wavelength-independent optical route for determining the twist angle in 2D materials. By linking the structural twist angle directly to the rotation of SHG polar lobes ($\theta_{tw} = 2\phi_0$), the authors provide a non-destructive, high-throughput tool for characterizing moiré superlattices and stacking domains. Furthermore, the work offers a clear roadmap for engineering nonlinear optical responses in 2D materials solely through stacking control, enabling the design of heterostructures with tailored functionalities such as complete SHG suppression or specific tensor component activation. The framework is presented as universal for 2H-TMD bilayers and analogous van der Waals crystals, providing a foundational basis for future high-throughput screening of nonlinear optical architectures.