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

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.

💡 Research Summary

This paper presents a comprehensive study of charged excitonic complexes (trions) in monolayer transition‑metal dichalcogenides (TMDCs) using the Gaussian Expansion Method (GEM), a variational technique originally developed for few‑body problems in nuclear physics. The authors adapt GEM to the two‑dimensional (2D) environment by employing the Rytova‑Keldysh (RK) potential to model the screened Coulomb interaction between carriers and by formulating the three‑body Hamiltonian in Jacobi coordinates. Three rearrangement channels are included, allowing the method to treat the mass asymmetry between electrons and holes naturally.

The basis set consists of non‑orthogonal Gaussian functions with widths generated by geometric progressions, providing dense coverage at short inter‑particle distances and sufficient extension to capture the long‑range tail of weakly bound states. For each Jacobi coordinate, 20 Gaussian functions are used (n_max = N_max = 20) with range parameters r₁ = R₁ = 0.1 Å and r_max = R_max = 200 Å. Angular momentum quantum numbers (ℓ, L) are truncated to |ℓ|,|L| ≤ 4, and the total orbital angular momentum J = ℓ + L is imposed.

The method is first benchmarked on the well‑studied J = 0 (s‑wave) negative trion (X⁻) in four representative TMDCs: MoS₂, MoSe₂, WS₂, and WSe₂. Binding energies obtained with GEM lie between 25 meV and 35 meV, in excellent agreement (within ~1 meV) with results from stochastic variational methods (SVM), diffusion Monte Carlo (DMC), hyperspherical harmonics (HH), and other approaches reported in the literature. Convergence tests show that adding higher‑order angular channels changes the total three‑body energy by less than 0.1 meV, confirming the robustness of the basis.

A key new finding is the identification of a bound J = 1 trion state. While the J = 0 ground state is dominated by the (ℓ, L) = (0, 0) configuration, the J = 1 state emerges primarily from the (ℓ, L) = (1, 0) and (0, 1) channels. The calculated binding energy of this excited trion is modest, around 1–1.3 meV, indicating a shallow but genuine bound state that has not been reported in previous theoretical works.

The authors further analyze the internal structure of both J = 0 and J = 1 trions by evaluating probability density distributions in Jacobi coordinates. The J = 0 trion exhibits a nearly isotropic triangular geometry with an average electron‑electron separation of ~1.5 nm and electron‑hole separation of ~0.9 nm. In contrast, the J = 1 trion shows a pronounced asymmetry: one electron is closer to the hole while the other remains farther away, leading to a distorted triangle. These structural differences are expected to affect optical selection rules, Zeeman splitting, and valley‑dependent dynamics.

Environmental effects are explored systematically. By varying the average dielectric constant ε_m (modeling different substrates) from 1 (suspended) to 4 (high‑κ dielectric) and applying biaxial strain of ±2 %, the authors find that the J = 1 binding energy is highly sensitive: increasing ε_m reduces the binding by ~30 %, while compressive strain enhances it by ~20 %. The J = 0 binding energy is less affected, reflecting its stronger Coulomb binding.

From a methodological standpoint, GEM offers several advantages over existing techniques. Because most matrix elements can be evaluated analytically, the computational cost is modest, and memory requirements are far lower than those of DMC or large‑scale diagonalization in momentum space. Moreover, the explicit form of the wave function facilitates direct interpretation of spatial correlations, which is cumbersome in stochastic approaches.

In conclusion, the paper demonstrates that the Gaussian Expansion Method is a powerful, accurate, and computationally efficient tool for tackling three‑body problems in 2D semiconductors. It reproduces known trion binding energies, predicts a novel J = 1 bound trion, and quantifies how substrate screening and strain can be used to engineer trion properties. The authors suggest that GEM can be readily extended to more complex few‑body excitonic complexes (e.g., biexcitons, charged biexcitons) and to anisotropic 2D materials such as phosphorene, opening new avenues for theoretical design of optoelectronic and quantum information devices based on layered materials.