Disorder of excitons and trions in monolayer MoSe₂

Transition metal dichalcogenide (TMDC) monolayers have emerged as the most versatile models for the exploration of many-body semiconductor physics in two-dimensions (2D). The interplay of 2D character and poorly screened Coulomb potential leads to strong many-body effects that dominate the optical properties of TMDCs. Strongly bound excitons, with binding energies in the hundreds of meV, and strongly bound trions, with binding energies in the tens of meV, have been observed in TMDCs.^1–4 These tightly bound excitonic complexes are attractive models for the understanding of many-body interactions in 2D and for optoelectronic applications. However, most 2D materials are strongly affected by disorder^5,6 whose sources include material defects,⁷ electrostatic potential fluctuations from variations in local charge distributions,^8,9 dielectric constant variations,¹⁰ and strain-induced bandgap changes.^11,12 Interestingly, as we discuss in this Communication, even state-of-the-art TMDC monolayers with hexagonal boron nitride (h-BN) encapsulation are not immune to disorder. The response of many-body exciton and trion states to disorder can provide valuable insights into both the nature of the disorder and the nature of the excitonic complexes. Here, we carry out hyperspectral imaging of excitons and trions in h-BN encapsulated monolayer MoSe₂. We find that both exciton and trion energies are sensitive to variations in the local environment, but these two energies are almost perfectly correlated, in contrast to the behavior of the energy gap between the two lowest exciton states. Statistical analysis of the spatial energy variations, combined with theoretical modeling of exciton and trion states in the presence of disorder, reveals that the sources of the disorder are dielectric constant and electronic bandgap variations of Δε ∼ 0.08 and ΔE_g ∼ 2–3 meV, respectively.

In experiments, we use the highest quality monolayers exfoliated from flux grown MoSe₂ single crystals with low defect density (∼10¹¹ cm⁻²), as quantified by scanning tunneling microscope (STM) imaging detailed recently,¹³ and large areas (>160 µm²). Each MoSe₂ monolayer is encapsulated in h-BN. The steady-state photoluminescence (PL) spectra [Fig. 1(a)] show two narrow peaks assigned to the lowest energy 1s exciton and the lowest energy trion,^3,4 with mean energies $E1sex$ = 1.6465 ± 0.0001 eV and E^tr = 1.6214 ± 0.0001 eV, respectively. The difference $E1sex−Etr$ is often referred to as the trion binding energy $Ebtr$⁠. We use MoSe₂ monolayers exfoliated from the highest quality single crystal, with quantified defect density of 8 ± 5 × 10¹⁰ cm⁻² from scanning tunneling microscopy (STM), with calibrated PL quantum yield of 70%, and with full-width-at-half-maximum (FWHM) of both exciton and trion PL peaks approaching ∼1 meV, as detailed recently,¹³ in agreement with previous reports.^7,14 We determine the effects of disorder^{7–12,15–18} by hyperspectral PL imaging with a spatial resolution of ∼1 µm. A continuous wave laser at hν = 2.33 eV excites the sample in a diffraction-limited spot of FWHM ∼ 0.43 µm, at 4.66 µW/µm², corresponding to a calibrated total exciton and trion density of ∼1 × 10¹⁰/cm².¹³ The high sample quality gives rise to relatively homogeneous spatial distributions in total PL intensity [Fig. 1(a), inset] and the trion-to-exciton intensity ratio I_tr/I_ex [Fig. S1(a)]. Note that we find no correlation between I_tr and I_ex [Fig. S1(b)], suggesting the absence of measurable electron density fluctuation,⁸ which would give rise to anti-correlation in I_tr and I_ex.

We now focus on $E1sex$ and E^tr, extracted for each spot from intensity-weighted averaging of the PL spectra [Fig. 1(a)]. While $E1sex$ and E^tr fluctuate over the whole sample area [Figs. 1(b) and 1(c)]. The difference between the two shows a surprisingly uniform spatial distribution with a mean value of $Ebtr$ = 26.220(1) ± 0.0005 meV [Fig. 1(d)]. This suggests that spatial fluctuations in $E1sex$ and E^tr are highly correlated. Figure 1(e) shows a scatter plot of E^tr vs $E1sex$⁠. The solid and dashed lines indicate the directions of the eigenvectors of the covariance matrix of $E1sex$ and E^tr with slopes of +0.97 and −1.03, respectively. Note that slopes of ±1 would indicate perfect correlation and slopes of 0 and ∞ would indicate no correlation. we plot histograms of $E1sex$⁠, E^tr, and $Ebtr$ in Figs. 1(f)–1(h), corresponding to standard deviations of $σ1sex$ = 3.6 ± 0.2 meV, σ^tr = 3.5 ± 0.2 meV, and $σbtr$ = 0.186 ± 0.001 meV, respectively. Because of the nearly perfect correlation between $E1sex$ and E^tr, $σbtr$ is only ∼5% of $σ1sex$ and σ^tr. The insensitivity of $Ebtr$ to disorder is seen in a broad temperature range until T ∼ 60 K, above which the trion PL peak disappears, likely attributed to dissociation of the many-body trion complex by phonon scattering (Figs. S2 and S3).

In stark contrast to the nearly constant $Ebtr$⁠, the energy splitting between the exciton levels is more broadly distributed. We quantify the spatial distributions in 1s and 2s exciton energies from reflectance contrast (R_c) spectra on the same sample as in Fig. 1. Figure 2(a) is a representative R_c spectrum showing the A-exciton 1s (1.649 eV) and 2s (1.799 eV, inset) transitions, and the B-exciton 1s transition (1.847 eV), with energies in agreement with previous reports.^10,19 The $E1sex$ and $E2sex$ energy maps for the A-1s exciton in Figs. 2(b) and 2(c) show characteristic spatial variations attributed to the disorder. The energy gap, $EΔex=E2sex−E1sex$⁠, also shows spatial variation of the same order [Fig. 2(d)]. A scatter plot of $E2sex$ vs $E1sex$ is shown in Fig. 2(e), along with solid and dashed blue lines showing the directions of the eigenvectors of their covariance matrix. The slopes, +1.44 and −0.69, indicate a much weaker correlation than that between $E1sex$ and E^tr. Figures 2(f)–2(h) show histograms of $E1sex$ and $E2sex$⁠, and $EΔex$ with standard deviations $σ1sex$ = 3.1 ± 0.2 meV, $σ2sex$ = 4.4 ± 0.4 meV, and $σΔex$ = 1.7 ± 0.1 meV, respectively. The weak correlation between $E1sex$ and $E2sex$ results in $σΔex$ being 40%–60% of $σ2sex$ and $σ1sex$⁠. Note that a comparison of Fig. 2(f) from reflectance and Fig. 1(f) from PL gives a small Stokes shift of −1.5 ± 01 meV, consistent with the low defect density of our sample. Note also that the spin–orbit splitting in the valence band, as reflected in the difference between A and B exciton energies, is an intrinsic property of the monolayer and is robust against spatial variations, Fig. S4.

We point out that the use of the state-of-the-art sample of the lowest defect density to date¹³ is key to the first quantitative comparison between experiment and theory on exciton and trion disorder in a TMDC monolayer. This comparison identifies excitonic disorder as resulting from dielectric fluctuation and strain-related bandgap changes, not electrostatic variation. While dielectric fluctuation may come from impurity charges and adsorbates,¹⁰ strain may have intrinsic origin due to atomic scale relaxation at interface,²⁴ and extrinsic sources from the transfer stacking process²⁵ or roughness of the substrate.⁵ Further minimizing disorder may necessitate the development of automated transfer stacking processes in ultra-clean environments. We note that previous spectroscopic efforts in exploring the disorder problem have focused on distinguishing homogeneous from heterogeneous linewidth of exciton transitions from coherent spectroscopies.^26–28 In a four-wave mixing experiment on exciton-trion coherence, Jakubczyk et al. also reported a nearly constant trion binding energy despite the presence of more extensive spatial disorder from the use of more defective samples without encapsulation; these authors noted the interesting observation without providing an interpretation.²⁸ 

In conclusion, we have compared the behaviors of excitons and trions in the presence of disorder in monolayer MoSe₂ encapsulated in h-BN. Hyperspectral imaging revealed that the 2s-1s exciton energy splitting varies by $σΔex$ = 1.7 ± 0.1 meV due to disorder. In contrast, the trion binding energy is robust with spatial variation of only $σbtr$ = 0.186 ± 0.001 meV, which is one order of magnitude lower than $σΔex$⁠. Theoretical analysis based on the many-body exciton-trion quantum superposition model²⁰ provides a quantitative explanation of the experimental results and suggests dielectric and strain origins, not an electrostatic one, for exciton and trion disorder.