Two-dimensional bound excitons in the real space and Landau quantization space: a comparative study

The Landau quantization space is based on the respective motion of the electron and hole in a magnetic field and can provide a new route to understand the bound exciton behaviors observed in the experiments. In this paper, we study the two-dimensional exciton properties of monolayer WSe$_2$ in both the real space and Landau quantization space. Focusing on the excitons of zero center-of-mass momentum, we calculate its energy spectrum in both spaces, with the results agreeing well with each other. We then obtain the diamagnetic coefficients and root-mean-square radius, which are consistent with the available $s$ state data in the experiment. More importantly, in the exciton state $nl$, we find that the dominant electron-hole pair component may shift with the magnetic field and the Coulomb interactions, and reveal that the magnetic field will drive the dominant component to be the free electron-hole pair ${n_e=n+l-1,n_h=n-1}$, whereas the Coulomb interactions drives it to be the pair of the lower index.

💡 Research Summary

This paper presents a comprehensive theoretical comparison of two‑dimensional (2D) bound excitons in monolayer WSe₂ using two complementary frameworks: the conventional real‑space approach and the Landau‑quantization‑space (LQS) formalism. The authors focus on excitons with zero center‑of‑mass momentum (K = 0), which are optically active, and calculate their energy spectra, diamagnetic coefficients, and root‑mean‑square (rms) radii in both representations.

In the real‑space treatment, the effective‑mass Hamiltonian incorporates the Keldysh screened Coulomb potential, which captures the non‑local dielectric environment of a 2D material. After a gauge transformation to the symmetric gauge, the relative‑motion Schrödinger equation reduces to a radial differential equation containing kinetic, Zeeman, and quadratic diamagnetic terms. Because the Keldysh potential lacks an analytic solution, the authors discretize the radial coordinate (Δr = 0.05 nm, r_cut = 60 nm) and convert the problem into a generalized eigenvalue equation AV = εBV, solved with LAPACK’s dggev routine.

The LQS approach starts from the Landau gauge (A = Bx ŷ) and expands the free electron and hole states in Landau level (LL) bases |nₑ, kᵧᵉ⟩ and |nₕ, kᵧʰ⟩. Interaction matrix elements are expressed through form factors S_{n′n}(q), which are analytically reduced to Laguerre‑polynomial forms, ensuring numerical stability even for high LL indices. The bound exciton is written as a superposition of free electron‑hole pairs, |Ψ_K⟩ = ∑_{nₑ,nₕ} ϕ_K(nₑ,nₕ) |nₑ,kₑ; nₕ,K−kₑ⟩, and the Coulomb kernel leads to a matrix eigenvalue problem (Eqs. 26‑27).

Both methods yield virtually identical exciton energy ladders over magnetic fields ranging from 0 to 30 T. For the 1s (n = 1, l = 0) state, the calculated diamagnetic coefficient σ_{1s}≈0.5 µeV T⁻² and rms radius r_{1s}≈1 nm agree with recent magneto‑photoluminescence measurements.

A key finding concerns the composition of an exciton state labeled nl. By analyzing the eigenvectors in the LQS basis, the authors discover that the dominant electron‑hole pair component shifts with magnetic field strength and with the strength of the screened Coulomb interaction (controlled by the dielectric environment). At high magnetic fields the “free” pair {nₑ = n + l − 1, nₕ = n − 1} becomes dominant, reflecting the tendency of the magnetic quantization to align the pair with the Landau ladder. Conversely, when the Coulomb interaction is strong (short screening length r₀ or high dielectric constant), the lower‑index pair {nₑ = n, nₕ = n + l} prevails. Phase diagrams in the (B, εᵥ) plane illustrate the competition between these two tendencies and delineate clear transition boundaries.

The authors discuss the physical implications: the magnetic‑field‑driven re‑configuration of the electron‑hole pair composition can modify optical selection rules, transition dipole moments, and polarization properties, offering a route to actively control excitonic resonances in 2D TMD devices. The agreement between the two theoretical frameworks validates the LQS formalism as a powerful complement to real‑space calculations, especially for problems where the Landau level structure plays a central role.

In conclusion, the paper demonstrates that real‑space and Landau‑quantization‑space approaches are quantitatively consistent for 2D excitons in WSe₂, while revealing a novel magnetic‑field‑induced shift in the internal electron‑hole pair makeup. This insight deepens our understanding of exciton physics in strongly screened 2D semiconductors and suggests new strategies for engineering excitonic properties via external magnetic fields and dielectric engineering. Future work may extend the analysis to anisotropic effective masses, many‑body exciton‑exciton interactions, and non‑uniform magnetic textures.