Gaussian Expansion Method for few-body states in two-dimensional materials

TL;DR

This work develops and applies the Gaussian Expansion Method (GEM) to study three-body trion states in two-dimensional TMDC monolayers, using the Rytova–Keldysh potential to model nonlocal screening. GEM treats the three-body wavefunction as a sum over rearrangement channels with a geometrically parameterized Gaussian basis, achieving rapid convergence and providing analytical matrix elements through the Infinitesimally Shifted Gaussian-Lobe formalism. The study finds a robust $J=0$ trion binding energy around $\sim$33 meV and uncovers a bound excited $J=1$ trion with $E_B$ between $0.39$ and $1.44$ meV across several TMDCs; it also analyzes the internal structure, density distributions, and trion geometry, and investigates how strain and dielectric environment influence these states. The results validate GEM against alternative methods (SVM, DMC, DVR, MSRE, HH) and offer insights into how environmental factors may tune weakly bound trions, pointing to extensions to multilayer and anisotropic materials and potential plasmon–exciton coupling applications.

Abstract

We investigate the properties of trions in transition metal dichalcogenides (TMDCs) monolayers using the Gaussian Expansion Method (GEM) adapted to two-dimensional systems. Excitons and trions in monolayer TMDCs with the chemical composition MX$_2$ in the 2H phase are studied systematically. We computed the associated exciton and trion binding energies. We find in addition to the known $J = 0$ trion the existence of a bound state with orbital angular momentum $J = 1$. The results for $J = 0$ are benchmarked against existing calculations from the Stochastic Variational Method (SVM) and Quantum Monte Carlo (QMC). Furthermore, we analyze the trion internal structure and geometry through their probability density distributions, accounting for the effects of different material shows that GEM -- widely used in studies of strongly interacting few-body systems -- is well adapted to allow comprehensive and computationally efficient investigations of trions and potentially other weakly bound few-body states in layered materials. In addition, we systematically exploit the effect of strain and dieletric environment in the $J = 1$ trion predictions, illustrated for the MoS$_2$ monolayer example.

Figures (7)

• Figure 1: Schematic representation of the trion $X^-$ and TMDCs. In the left panel (a), we illustrate the trion in a single layer TMDC of the form MX$_2$, namely MoS$_2$, MoSe$_2$, Ws$_2$, and WSe$_2$. In our study the electrons are in a spin-singlet configuration as presented in the right panel (b), where the dashed lines represent the spin-down band and the full line represents spin-up band, for the conduction band (CB, blue) and valence band (VB, red).
• Figure 2: Jacobi coordinates for the three-body system of two electrons and one hole. $\mathbf{r}_c$ denotes the relative coordinates of the two-body subsystem and $\mathbf{R}_c$ labels the spectator particle position relative to the center of mass of the two-body subsystem. The label $c$ indicates the set of Jacobi coordinates.
• Figure 3: Trion $J=0$ and $J=1$ density distributions for a single layer of MoS$_2$ and defined in Eq. \ref{['eq:densrR']}. Top panels (a) and (b) show the densities for $J=0$ state. Bottom panels (c) and (d) show the densities for $J=1$ state. The solid curves indicate the Jacobi coordinates for set 1 represented in panel (e) and the dashed curves label set 2 in panel (f). The parameters used to compute the trion in MoS$_2$ are $m_e = m_h = 0.5\,m_0$ for the effective masses and $r_0 = 41.469$ Å for the screening length.
• Figure 4: Trion geometry in MoS$_2$. Panel (a) corresponds to $J=0$ and panel (b) to $J=1$ states, respectively. These figures are obtained with $m_e = m_h = 0.5 m_0$ and $r_0 = 41.469$ Å.
• Figure 5: Trion binding energy landscape for $J=1$ (top panel) and $J=0$ (bottom panel) states with variation in polarizability and charge carrier mass ($m_e=m_h$) in vacuum. The circles represents the different TMDs given in Table \ref{['tab:Table_Comp']}. The box at the center of the figures estimate the MoS$_2$ trion binding energies for small biaxial strain variations between 0% and 2.0% (see explanations in the text and Table \ref{['tab:model_params']}).