Many-Body effects beyond excitons in second-harmonic generation of monolayer MoS$_{2}$

Abstract

We present a quantitative study of many-body effects including the three-particle level on second-harmonic generation in monolayer MoS$_{2}$. Our approach combines many-body perturbation theory with time-dependent current-density-functional theory within an \textit{ab initio} framework in the optical limit. Inclusion of two-particle excitonic effects \textit{via} a dynamical long-range linear exchange-correlation kernel reproduces the qualitative features of the second-harmonic response, but underestimates the experimentally reported magnitudes by nearly a factor of two. By incorporating three-particle (trionic) correlations through a static long-range quadratic exchange-correlation kernel, we achieve significantly improved quantitative agreement with experiment. These findings highlight the role of many-body interactions beyond the excitonic level in accurately describing second-order optical responses in two-dimensional semiconductors.

Figures (5)

• Figure 1: Schematic of two-particle (a) and three-particle (b,c) many-body interactions in the linear and quadratic optical response of a semiconductor, respectively. Grey and white shaded lines represent valence and conduction bands, respectively.
• Figure 2: (a) Out-of-plane extension of the in-plane averaged charge density (left) together with the $OYZ$ side view of the slab crystal unit (right) of monolayer MoS$_{2}$. Supercell thickness, effective thickness, and half of the bulk thickness are $L_{\mathrm{sc}}=20~\mathrm{\AA}$, $L_{\mathrm{eff}}=7.3~\mathrm{\AA}$ and $L_{c/2}=6.15~\mathrm{\AA}$young, respectively. The green-shaded area highlights the effective out-of-plane spatial extent of the electrons. (b) LDA (solid grey lines) and $G_{0}W_{0}$ (dashed red lines) electronic band structure of monolayer MoS$_{2}$. LDA and $G_{0}W_{0}$ direct energy band gaps are equal to $1.65~\mathrm{eV}$ and $2.52~\mathrm{eV}$, respectively.
• Figure 3: (a) Macroscopic optical dielectric spectrum of MoS$_{2}$. Dashed and solid yellow lines with circles represent the real and imaginary parts of experimental data, respectively, from Ref. Ermolaev2020. Dashed and solid narrow red (broad blue) lines respresent the real and imaginary parts within the $G_{0}W_{0}$ (BSE) picture, respectively. (b) In-plane longitudinal component of the linear LRC screeening tensor $\bm{\alpha}_{\mathrm{LRC}}(\omega)$. Solid and dashed green lines represent the real and imaginary parts, respectively.
• Figure 4: Absolute value of the macroscopic two-dimensional SHG susceptibility per unit area of monolayer MoS$_{2}$. The yellow line with circles dots represents experimental data from Ref. PhysRevB.87.201401. Blue lines with diamonds and green lines with squares represent theoretical results including excitonic effects from Refs. PhysRevB.89.081102PhysRevB.90.199901 and PhysRevB.89.235410, respectively. Red lines represent our theoretical results including excitonic effects.
• Figure 5: (a) Absolute value of the macroscopic two-dimensional SHG susceptibility per unit area of monolayer MoS$_{2}$. Yellow dots represent experimental data from Ref. PhysRevB.87.201401. Red lines represent our theoretical results including many-body interactions up to the two-particle level, i.e. excitonic effects, while dashed blue line includes also at the three-particle level, i.e. trionic effects. (b) Theoretical value of the height of the peak close to $1.5~\mathrm{eV}$ as a function of the in-plane longitunal component of the quadratic LRC $\bm{\beta}_{\mathrm{LRC}}$ tensor of the quadratic exchange-correlation kernel with the dotted line being the experimental value.