[Paper Review] Exciton band structure of monolayer MoS2

Fengcheng Wu, Fanyao Qu+1 more|arXiv (Cornell University)|Jan 9, 2015

2D Materials and Applications17 citations

TL;DR

This theoretical study reveals that monolayer MoS2 exhibits complex excitonic behavior due to strong electron-hole interactions and broken inversion symmetry, with excitons existing at both the Brillouin zone center and corners. The key finding is that despite being a direct-gap semiconductor in band structure, MoS2 behaves as an indirect-gap material in its excitation spectra due to low-energy excitons at the Brillouin-zone corners, and finite-momentum excitons split into a quadratic and a non-analytic linear dispersion mode due to valley coherence from exchange interactions.

ABSTRACT

We address the properties of excitons in monolayer MoS$_2$ from a theoretical point of view, showing that low-energy excitonic states occur both at the Brillouin zone center and at the Brillouin-zone corners, that binding energies at the Brillouin-zone center deviate strongly from the $(n-1/2)^{-2}$ pattern of the two-dimensional hydrogenic model, and that the valley-degenerate exciton doublet at the Brillouin-zone center splits at finite momentum into an upper mode with non-analytic linear dispersion and a lower mode with quadratic dispersion. Although monolayer MoS$_2$ is a direct-gap semiconductor when classified by its quasiparticle band structure, it may well be an indirect gap material when classified by its excitation spectra.

Motivation & Objective

• To understand the full momentum-space exciton band structure in monolayer MoS2 beyond the zone-center approximation.
• To investigate how broken inversion symmetry and valley degrees of freedom influence excitonic states and their optical properties.
• To determine whether monolayer MoS2 should be classified as a direct or indirect gap semiconductor based on its excitation spectra.
• To analyze the role of electron-hole exchange interactions in splitting valley-degenerate excitons at finite momentum.
• To compare theoretical predictions with the 2D hydrogenic model and identify deviations due to band structure and screening effects.

Proposed method

• Employed a five-band d-orbital tight-binding model to describe quasiparticle bands of monolayer MoS2, including spin-orbit coupling.
• Used the Bethe-Salpeter equation (BSE) to solve for excitonic states, incorporating momentum-dependent electron-hole interactions and form factors.
• Applied a massive Dirac model with a screened Coulomb interaction (via form factor F(q) = 1/(1 + r₀q)) to analytically study binding energies.
• Calculated optical conductivity to probe zero-center-of-mass momentum excitons and their splitting due to spin-orbit coupling.
• Performed numerical solutions of the BSE on a 45×45 k-grid to map exciton dispersions across the Brillouin zone.
• Used effective atomic units and an ansatz ψ(k) = ψ(k)e^{ilϕk} to reduce the 2D BSE to a 1D eigenvalue problem for numerical solution.

Experimental results

Research questions

• RQ1 How do excitonic states in monolayer MoS2 disperse across the entire Brillouin zone, particularly at finite momentum?
• RQ2 Why do binding energies in monolayer MoS2 deviate from the (n−1/2)⁻² scaling of the 2D hydrogenic model?
• RQ3 What is the origin of the splitting between the 2s and 2p states in the A-series excitons, and why are 2p states more bound than 2s?
• RQ4 How do valley coherence and exchange interactions affect the dispersion of finite-momentum excitons?
• RQ5 Is monolayer MoS2 a direct or indirect gap semiconductor when judged by its excitation spectra rather than quasiparticle band structure?

Key findings

• Low-energy excitonic states exist not only at the Brillouin zone center but also at the corners, indicating a complex excitation spectrum.
• Binding energies at the zone center deviate significantly from the 2D hydrogenic model: 2p states are lower in energy than 2s states, and the 2p states are split due to valley-dependent exchange interactions.
• Finite-momentum excitons split into two modes: a lower mode with quadratic dispersion and an upper mode with non-analytic linear dispersion, due to valley coherence from exchange interactions.
• The A-series excitons at Q=0 show that 2p states have larger binding energies than 2s states, contrary to the hydrogenic model, due to the specific band structure and screening.
• Despite being a direct-gap semiconductor in band structure, monolayer MoS2 may be an indirect-gap material in its excitation spectra due to the presence of low-energy excitons at the Brillouin-zone corners.
• The massive Dirac model with screened Coulomb interaction reproduces key features of the lattice BSE results, validating the theoretical framework.