Minimal d-Band Model for the Optical Susceptibility of Non-Centrosymmetric Monolayer Transition Metal Dichalcogenides

Technical Summary: Minimal d-Band Model for the Optical Susceptibility of Noncentrosymmetric Monolayer Transition Metal Dichalcogenides

Problem Statement
The optical response of two-dimensional materials, particularly monolayer transition metal dichalcogenides (TMDCs), is traditionally calculated ab initio using plane-wave basis sets and full Bloch wavefunctions. While accurate, these methods do not explicitly separate the most significant orbital contributions, often obscuring the physical mechanisms driving the response. Furthermore, first-principles calculations for nonlinear optical properties are computationally expensive, making the inclusion of many-body effects (such as excitonic correlations) challenging. In noncentrosymmetric monolayer TMDCs (2H stacking), the energy bands near the semiconductor gap are dominated by the $d$-orbitals of the transition metal atoms, with negligible contributions from the chalcogen $p$-orbitals. This observation raises the question of whether a minimal orbital basis model can accurately reproduce the linear and quadratic optical susceptibilities of these materials without resorting to full ab initio calculations.

Methodology
The authors develop a minimal three-band model based on the transition metal $d$-orbitals ($d_{z^2}$, $d_{xy}$, and $d_{x^2-y^2}$) to calculate the linear ($\chi^{(1)}$) and second-order ($\chi^{(2)}$) optical susceptibilities of monolayer TMDCs, using tungsten disulfide (WS$_2$) as a prototype. The approach operates within the single-particle approximation, disregarding many-body effects initially, but is designed to serve as a foundation for their future inclusion.

Key methodological steps include:

1. Hamiltonian Construction: The model utilizes a tight-binding (TB) Hamiltonian with third-nearest neighbor (TNN) interactions, building upon the symmetry-based three-band model introduced by Liu et al. This ensures the energy bands match ab initio results across the entire Brillouin zone (BZ), not just near the gap edges.
2. Symmetry Reduction: Leveraging the $D_{3h}$ point group symmetry and time-reversal symmetry, the authors reduce the integration domain from the full BZ to the time-reversed irreducible Brillouin zone (tIBZ). This significantly lowers the computational cost.
3. Momentum Matrix Elements: Instead of using full Kohn-Sham orbitals, the momentum matrix elements are constructed using pseudo-atomic orbitals (PAOs) derived from Density Functional Theory (DFT). The authors calculate two-center integrals of the momentum operator between these localized orbitals. The radial parts of the $d$-orbitals are fitted to Gaussian functions to facilitate numerical integration.
4. Spin-Orbit Coupling (SOC): SOC is included approximately. The authors note that while SOC significantly splits the valence and conduction bands, its direct effect on the velocity operator (anomalous velocity) is negligible (<1%) for the optical response in this context. Thus, the model incorporates SOC primarily through the modification of energy bands and eigenstates.
5. Formalism: The linear and second-order susceptibilities are derived using the density matrix perturbation theory in the long-wavelength approximation. The authors explicitly derive the non-zero tensor components allowed by $D_{3h}$ symmetry, showing that for linear susceptibility, $\chi^{(1)}_{xx} = \chi^{(1)}_{yy}$, and for second-order susceptibility, $\chi^{(2)}_{xxy} = \chi^{(2)}_{yxx} = -\chi^{(2)}_{yyy}$.

Key Contributions and Results

• Validation of the Minimal Model: The primary result is that the three-band $d$-orbital model reproduces first-principles calculations for optical susceptibility remarkably well. The model matches ab initio results for photon energies up to approximately 1.7 eV above the band gap (roughly 2 eV total).
• Efficiency: By reducing the calculation to a minimal basis set and exploiting symmetry to restrict integration to the tIBZ, the method offers a numerically inexpensive alternative to full plane-wave DFT calculations.
• Symmetry Analysis: The paper provides a rigorous derivation of how $D_{3h}$ symmetry and time-reversal symmetry constrain the optical susceptibility tensors, reducing the number of independent components and the integration volume required for numerical evaluation.
• Integration Procedure: The authors detail a broadening method using weighted Gaussian functions to handle the Dirac delta functions in the susceptibility integrals, ensuring smooth dispersion relations.

Significance and Claims
The paper claims that this minimal approach is a viable starting point for incorporating many-body effects into optical calculations. Because the model relies on a small number of energy bands, it is computationally feasible to apply techniques based on tight-binding Hamiltonians (such as solving the Bethe-Salpeter equation) to include electron-electron correlations and excitonic effects, which are crucial in low-dimensional TMDCs but difficult to treat with full ab initio methods. The authors emphasize that while the current work focuses on the single-particle response, the simplicity of the model allows for the efficient addition of these complex many-body corrections. The study confirms that the $d$-orbital character of the bands near the gap is sufficient to capture the essential physics of the optical response in noncentrosymmetric monolayer TMDCs.