Coulomb Interaction in Atomically Thin Semiconductors and Density-Independent Exciton-Scattering Processes

TL;DR

The work develops a comprehensive, second-quantized treatment of Coulomb interactions in atomically thin semiconductors to enable quantum-kinetic simulations of excitons and higher-order correlations. It formulates a memory-dependent, screened Coulomb Hamiltonian that includes Umklapp and local-field effects, and links ab initio dielectric screening to few-band effective models via a microscopic dielectric function $\varepsilon_{\text{mic}}$ and a layered macroscopic approach. Exciton physics is treated through the Bethe-Salpeter equation in COHSEX and the Wannier equation, with explicit consideration of direct and exchange Coulomb scattering, including Dexter- and Förster-type processes in layered dielectrics. The framework also provides analytical screening models for 2D materials and a practical recipe to incorporate environmental screening, making it suitable for predicting exciton energies and density-independent scattering in TMDCs and related 2D semiconductors.

Abstract

In quantum-kinetic approaches to the dynamics of Coulomb-bound many-body correlations such as excitons, trions, biexcitons or higher-order correlations, a detailed knowledge of the many-body Coulomb Hamiltonian serving as a starting point is important. In this manuscript, the second-quantized description of the Coulomb interaction between Bloch electrons in a Heisenberg-equation-of-motion approach in atomically thin semiconductors is derived and reviewed. Emphasis is put on a discussion of Umklapp processes and the dielectric screening including all local-field effects. A link between \textit{ab initio} methods of screening and few-band models in effective-mass approximations for the quantum kinetics is established and all important Coulomb scattering processes contributing to the exciton energy landscape and density-independent exciton scattering are discussed.

Figures (14)

• Figure 1: Normal (a) and Umklapp (b) processes for a vanishing reciprocal lattice vector $\mathbf G^{\prime\prime}$ in an example square lattice.
• Figure 2: Scheme of the Coulomb scattering processes described by Eq. (\ref{['eq:Coulomb_Hamiltonian_SecondQuantized_Formfactors_Evaluated']}).
• Figure 3: Five dielectric volumes: The grey-colored volume represents the semiconductor monolayer $\epsilon_{\text{M}}$ with thickness $d$, red and blue denote superstrate $\epsilon_1$ and substrate $\epsilon_2$, respectively, with a small vacuum gap $h$.
• Figure 4: (a) The two-dimensional dielectric function $\epsilon_{\text{M}, \mathbf q_{\parallel}^{\newline}}^{\text{2D}}$ from Eq. (\ref{['eq:effective_confined_screening']}) of an example MoS$_2$ monolayer ($\epsilon_{\text{M},\parallel}=15.5$, $\epsilon_{\text{M},{\bot}}=6.2$, $\epsilon_{\text{M},\text{bulk}}=10.5$laturia2018dielectric, $d=0.618\,$nm kylanpaa2015binding, $\hbar\omega_{\text{pl}}=22.5\,$eV kumar2012tunable) suspended in vacuum ($\epsilon_1=\epsilon_2=1$) is shown. "$\epsilon_{\text{M}, \mathbf q_{\parallel}^{\newline}}^{\text{2D}}(\epsilon_{\text{M},\mathbf 0}^{\text{3D}})$" uses a constant approximation of the three-dimensional material background dielectric function $\epsilon_{\text{M},\mathbf 0}^{\text{3D}}$ and "$\epsilon_{\text{M}, \mathbf q_{\parallel}^{\newline}}^{\text{2D}}(\epsilon_{\text{M}, \mathbf q_{\parallel}^{\newline}}^{\text{3D}})$" uses the $\mathbf q_{\parallel}^{\newline}$-dependent model from Eq. (\ref{['eq:dielectric_function_material_q_dependent']}). "$\epsilon_{\text{M}, \mathbf q_{\parallel}^{\newline}}^{\text{R-K}}$" denotes the Rytova-Keldysh thin-film limit in vacuum from Eq. (\ref{['eq:epsilon_rytova_keldysh']}) and "CMR" denotes ab initio calculations extracted from the Computational Materials Repository (CMR) andersen2015dielectric. $\mathbf K^K$ is a $K$ valley momentum, cf. Sec. \ref{['sec:expansion_band_extrema']}, and $\mathbf G_{\parallel}^{\newline}$ is a reciprocal lattice vector. (b) The two-dimensional dielectric function $\epsilon_{\text{M}, \mathbf q_{\parallel}^{\newline}}^{\text{2D}}$ from Eq. (\ref{['eq:effective_confined_screening']}) for three different layer thicknesses and the three-dimensional dielectric function $\epsilon_{\text{M}, \mathbf q_{\parallel}^{\newline}}^{\text{3D}}$ from Eq. (\ref{['eq:dielectric_function_material_q_dependent']}) are shown.
• Figure 5: Fully occupied valence band (a), optical excitation with Rabi frequency $\Omega_{\sigma_+}$ in $\sigma_+$-polarization creates an electron-hole pair at the $K$ valley (b) article:THEORY_spin_valley_selective_excitation_Yao2012, which binds via electron-hole Coulomb attraction $K_{\text{eh}}^{\text{dir}}\newline_{ \mathbf q_{\parallel}^{\newline}}^{K,K}$, cf. Eq. (\ref{['eq:electron_hole_kernel_direct_reduced']}), forming an exciton at zero center-of-mass momentum ${{}\hat{P}^{\newline}_{}}{\newline}_{\mu,\mathbf 0}^{K,K,\uparrow,\uparrow}$ (c).