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

TL;DR

The paper addresses computing linear and nonlinear optical susceptibilities in non-centrosymmetric TMDC monolayers, whose low-energy bands are predominantly $d$-orbital in character. It develops a minimal three-band $d$-orbital tight-binding model for WS$_2$ in the 2H stacking ($D_{3h}$ symmetry) and derives $\chi^{(1)}$ and $\chi^{(2)}$ within a single-particle framework, exploiting symmetry to confine integrations to the time-reversed irreducible Brillouin zone and to evaluate two-center momentum integrals efficiently. The results show that this minimal model reproduces plane-wave ab initio optical responses up to about $1.7$ eV above the band gap, with the nonzero components constrained by symmetry and the dominant second-order component being $\chi^{(2)}_{xxy}$ (and related). This provides a computationally inexpensive foundation for incorporating many-body effects and spin-orbit coupling with only a small number of bands, enabling scalable studies of nonlinear optics in TMDCs.

Abstract

The optical response of two-dimensional (2D) materials has been customarily calculated ab initio using plane waves and without separating the most important orbitals contributions. In the family of transition metal dichalcogenides (TMDC) monolayers lacking inversion symmetry, we take advantage of the mostly d-orbital content of the Bloch bands around the semiconductor gap to reduce the calculation of the linear and quadratic optical susceptibilities to a very minimal model. Such a simple approach reproduces well first principles calculations and could be the starting point for the inclusion of many-body effects and spin-orbit coupling (SOC) in TMDCs with only a few energy bands in a numerically inexpensive way.

Figures (4)

• Figure 1: (a) Top view of monolayer WS$_2$ with trigonal prismatic configuration $2H$ and lattice vectors $\bm{a}_1$ and $\bm{a}_2$ (lattice parameter $a=|\bm{a}_1|=|\bm{a}_2|$). Sulfur layers lie one on top of the other. (b) Trigonal unit cell of WS$_2$ exhibiting its $D_{3h}$ symmetry. (c) Hexagonal first Brillouin zone (BZ), with the irreducible Brillouin zone (IBF, gray shaded region) and time-reversed irreducible region (tIBZ, hatched area). Regions $(2)$ through $(6)$ are obtained from the IBZ by applying the symmetry operations indicated.
• Figure 2: Radial part of the $d$ pseudo orbitals for tungsten. The dots are the DFT data while the solid line is the fitting in Eq. \ref{['fit']} ($a_0$ is the Bohr radius).
• Figure 3: (a) Energy bands of monolayer WS$_2$ in the nearest (NN, dashed line) and third nearest (TNN, solid line) neighbors approximation with TB parameters from Liu2013, along the path $\Gamma\rightarrow M\rightarrow K\rightarrow\Gamma$ in the BZ. (b) Density of states corresponding to the bands in (a), calculated using the 2D linear tetrahedral method.
• Figure 4: (a) Imaginary and real parts of the $\chi^{ (1)}_{xx}$ component of monolayer WS$_2$ as a function of the photon energy $\hbar\omega$. (b) Same as in (a) but for the $\chi^{ (2)}_{xxy}$ of the nonlinear susceptibility. For $\chi^{ (2)}_{xxy}$, photon energies up to 2.3 eV has been included only. (c) Logarithm of the energy difference $\text{ln}|\epsilon_{31}-2\epsilon_{21}|$ showing that it vanishes in a almost circular ring around $M$ point.