Theory of In-Plane-Magnetic-Field-Dependent Excitonic Spectra in Atomically Thin Semiconductors

TL;DR

This work develops a fully analytic theory for the in-plane magnetic-field–dependent excitonic spectra in atomically thin TMDCs, showing that spin-bright and spin-dark excitons hybridize and that dark excitons brighten in absorption. Using a Maxwell–Bloch framework with non-Hermitian dissipation, the authors derive explicit expressions for hybridized exciton energies $\hbar\omega_{x,B_{\parallel}}^{S}$, linewidths $\hbar\gamma_{B_{\parallel}}^{S}$, and the two-peak absorption spectrum $\alpha(\omega)$, including mixing coefficients $P_{B_{\parallel}}^{S}$ and interference terms. The theory is applied to MoSe$_2$ and MoS$_2$ (h-BN encapsulated) to reveal distinct field-dependent behaviors: MoSe$_2$ shows non-monotonic amplitude evolution due to comparable dark-bright splitting, linewidths, and coupling, while MoS$_2$ exhibits a more straightforward monotonic increase in peak strength. These results provide a practical framework for analyzing magneto-optical measurements and interpreting dark-bright mixing in TMDC excitons across different material regimes.

Abstract

The linear absorption spectrum of excitons in TMDC monolayers under the influence of an in-plane magnetic field is theoretically studied. We demonstrate that in-plane magnetic fields induce a hybridization between spin-bright and spin-dark exciton transitions, resulting in a brightening of spin-dark excitons. We analytically investigate spectral features including resonance energy shifts, broadening and amplitudes ratios. In particular, for a MoSe$_2$ monolayer with radiatively-limited linewidth, we find a complex interplay of dark-bright splitting and linewidth difference of both involved spin-bright and spin-dark excitons.

Figures (7)

• Figure 1: (a) Spin-bright (light-gray ellipse) and spin-dark (dark-gray ellipse) excitonic transitions of the A-exciton considered in our model at the $K$-valleys. (b) Corresponding spin-bright and spin-dark transitions of the B-exciton. (c): An optical field $E_0^{\sigma}$ induces a polarization $P^{\sigma}$ in a TMDC monolayer in the presence of an in-plane magnetic field $B_{\parallel}$.
• Figure 2: Real- and imaginary part of mixing coefficients $P_{B_\parallel}^{\mathcal{S}}$ for $\nu = 1s$ under influence of the in-plane magnetic field for MoSe$_2$ ($\hbar \gamma_{\text{nrad}} = 0.5$ meV).
• Figure 3: Linear absorption spectrum under influence of an in-plane magnetic field for $\nu = 1s$ with different field-strengths for the material MoSe$_2$ in (a)--(c) and MoS$_2$ in (d)--(f) encapsulated in h-BN with $\sigma_+$-polarised light for increasing non-radiative linewidths $\gamma_{\text{nrad}}$.
• Figure 4: Hybridized excitonic energies $\hbar \omega_{x,B_\parallel}^\mathcal{S}$ from \ref{['eq: magnetic frequency']} for MoSe$_2$ in \ref{['fig: energies MoSe2']} and MoS$_2$ in \ref{['fig: energies MoS2']} under influence of an in-plane magnetic field for increasing linewidth difference $\kappa$ from \ref{['eq: linewidth difference']}. The colourgradient of the curves show the values of the real parts of mixing coefficients from \ref{['eq: mixing coefficient']}, i.e. the hybridization coefficients. Red (blue) represents a spin-up (spin-down) electron-state and yellow denotes the degree of spin-mixing.
• Figure 5: Hybridized linewidth $\hbar \gamma_{ B_\parallel}^\mathcal{S}$, normalized w.r.t. summed linewidth $\hbar \gamma^b + \hbar \gamma^d$, in dependence of an in-plane magnetic field for MoSe$_2$ with $\kappa$ = 0.71 meV and $\hbar \gamma_{\text{nrad}} = 0.1$ meV. The colourgradient of the curves show the values of the real parts of mixing coefficients from \ref{['eq: mixing coefficient']}, i.e. the hybridization coefficients. Red (blue) represents a spin-up (spin-down) electron-state and yellow denotes the degree of spin-mixing.