ARPES signatures of trions in van der Waals materials

TL;DR

This work addresses the yet unresolved ARPES fingerprints of charged excitons (trions) in doped two-dimensional semiconductors. It develops a material-specific microscopic framework that solves Wannier-like equations for excitons and trions, transforms to the exciton/trion basis, and computes ARPES intensities via Fermi's golden rule. In n-doped $WSe_2$ monolayers, the trion resonance appears at one electron–exciton binding energy $\Delta E^T_e$ below the conduction-band minimum $E_c$, and the exciton peak lies at one exciton binding energy $\varepsilon^{}_{b,X}$ below $E_c$, with the trion signal exhibiting a flatter dispersion due to the heavy residual exciton mass $M_X$. For mass-imbalanced trions, a characteristic double-peak ARPES feature emerges from distinct binding to the residual exciton, and a thermally populated multiplet of trion states broadens the spectrum into multiple resonances; together these results provide a quantitative framework for identifying trions in ARPES and studying many-body Coulomb complexes in doped 2D semiconductors.

Abstract

Angle-resolved photoemission spectroscopy (ARPES) has recently emerged as a direct probe of excitonic correlations in two-dimensional semiconductors, resolving their dispersion and dynamics in energy-momentum space, including dark exciton states inaccessible to optical techniques. However, the ARPES fingerprint of charged excitons (trions), which plays a key role in all doped and gated 2D material systems, has remained unknown so far. We present a first theoretical analysis of trion signatures in monolayer transition-metal dichalcogenides, highlighting how the additional charge carrier modifies the spectral position and shape relative to neutral excitons in ARPES spectra. Interestingly, we further predict that mass-imbalanced trions yield a characteristic double-peak structure, clearly separated in energy and line shape from neutral excitons. The predicted temperature dependence of these features offers guidance for experimental investigations aimed at identifying trionic states, thereby establishing a framework for ARPES studies of many-body Coulomb complexes in doped two-dimensional semiconductors.

Figures (7)

• Figure 1: Schematic illustration of exciton and trion contributions to the ARPES spectrum of an n-doped semiconductor. After exciton/trion formation, the photoemission pulse (orange wavy line) breaks the quasiparticle (electron–hole or electron–electron–hole complex), ejecting one electron and leaving behind a hole/exciton. Trions and excitons are denoted by T$_{\rm{h, e_1, e_2}}$ and X$_{\rm{h, e_{1/2}}}$, respectively, with the subindices describing the constituent electrons ($e_{1/2}$) and holes ($h$). The thickness of the lines representing electron orbits and excitons indicates the relative binding strength with thicker lines denoting a stronger binding. (a) Exciton case: the photoemitted electron is ejected from the material and its signal in ARPES is located one exciton binding energy ($\varepsilon^{}_{b,X}$) below the conduction band minimum. (b) Trion case: in the case of mass-balanced trions (left panel), the two electrons have a similar effective mass, so emission of either yields the same final configuration, producing a single spectral feature in ARPES located one electron-exciton binding energy ($\Delta$E$^T_e$) below the conduction band minimum. In the case of mass-imbalanced trions (right panel), electrons with different effective masses have unequal binding energies to the residual exciton ($\Delta$E$_{e_{1/2}}^T$), leading to a characteristic double-peak structure in ARPES spectra.
• Figure 2: ARPES intensity $I(k_x,E)$ for an n-doped WSe$_2$ monolayer along a momentum cut in the $k_x$ direction around the $K^{(\prime)}$ valley at $T=10$ K, shown as a function of energy for the lowest (a) exciton state X$_{\mathrm{KK}^{\prime}}$ and (b) trion state T$_{\mathrm{K}^{\uparrow}\mathrm{K}^{\downarrow}\mathrm{K}^{\prime\uparrow}}$. In both cases, the signal appears at the valley of the ejected electron, but its spectral position and shape differ. The exciton-related signal lies one exciton binding energy $\varepsilon^{}_{X,b}$ below the conduction-band minimum and follows the valence-band curvature (set by the hole effective mass $m_h$), whereas the trion signal is shifted by the much smaller electron–exciton binding energy $\Delta E^T_e$ and exhibits a nearly flat dispersion, reflecting the larger effective mass of the residual exciton ($M_X$). We take the valence-band maximum $\mathrm{E}_{v}$ as the reference energy level. (c,d) Energy landscape of the lowest-lying (c) excitonic and (d) trionic states as a function of their center-of-mass momentum $\mathbf{Q}_{X,T}$, with energies measured relative to the corresponding exciton or trion ground state $E^0_{X/T}$ at $\mathbf{Q}=0$ (for each valley configuration).
• Figure 3: Room temperature ARPES intensity for a n-doped WSe$_2$ monolayer considering a thermalized Boltzmann distribution of the three energetically lowest trion states, T$_{\mathrm{K}^{\uparrow}\mathrm{K}^{\downarrow}\mathrm{K}^{\prime\uparrow}}$, T$_{\mathrm{K}^{\uparrow}\mathrm{K}^{\prime\uparrow}\mathrm{\Lambda}^{\uparrow}}$, and T$_{\mathrm{K}^{\uparrow}\mathrm{\Lambda}^{\uparrow}\mathrm{\Lambda}^{\prime\downarrow}}$. Panels (a) and (b) show the signal around the K$^{(\prime)}$ and the $\Lambda^{(\prime)}$ valley, respectively, relative to the conduction-band minimum (solid black line). For the mass-imbalanced trion T$_{\mathrm{K}^{\uparrow}\mathrm{K}^{\prime\uparrow}\mathrm{\Lambda}^{\uparrow}}$, the origin of the double signal is indicated by orange boxes around the corresponding ejected electron. (c) Momentum-integrated ARPES signal illustrating the emergence of multiple trionic resonances and the pronounced spectral broadening compared to the low-temperature case (Fig. \ref{['fig:exciton_vs_trion']}), reflecting the broader center-of-mass momentum distribution at room temperature.
• Figure 4: Temperature dependence of the K and $\Lambda$ valley momentum-integrated ARPES signal $I(E)$ for a WSe$_2$ monolayer in the case of (a) excitons and (b) trions. For excitons, the low-energy states include X$_{\mathrm{KK}^{\prime}}$ and X$_{\mathrm{K}\Lambda}$, while for trions the relevant states are T$_{\mathrm{K}^{\uparrow}\mathrm{K}^{\downarrow}\mathrm{K}^{\prime\uparrow}}$, T$_{\mathrm{K}^{\uparrow}\mathrm{K}^{\prime\uparrow}\mathrm{\Lambda}^{\uparrow}}$, and T$_{\mathrm{K}^{\uparrow}\mathrm{\Lambda}^{\uparrow}\mathrm{\Lambda}^{\prime\downarrow}}$, see Figs. \ref{['fig:exciton_vs_trion']}c-d.
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