Optical fingerprint of bright and dark localized excitonic states in atomically thin 2D materials

Ermin Malic; Maja Feierabend; Samuel Brem

REVIEW 3 major objections 6 minor 65 references

Optical fingerprint of bright and dark localized excitonic states in atomically thin 2D materials

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Monolayer WSe2 with a Gaussian disorder potential should emit two families of localized exciton peaks below the bright line, with a temperature-controlled crossover near 20 K.

desk verdict A solid extension of the authors' PL framework to disorder-localized excitons, with interesting predictions for dark sidebands, but the 1s truncation and parameter fitting leave the key crossover under-validated. read the letter →

arxiv 1908.05071 v1 pith:IRHZDZ7Z submitted 2019-08-14 cond-mat.mes-hall physics.optics

Maja Feierabend , Samuel Brem , Ermin Malic This is my paper · ORCID

classification cond-mat.mes-hallphysics.optics

keywords localizedexcitonstransitionmetaldichalcogenidesphotoluminescencedarkdisorderpotentialexciton-phononscatteringWSe2single-photonemission

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper predicts that at low temperatures a monolayer of WSe$_2$ with a Gaussian-shaped disorder potential—from local strain, a nanopillar, or an impurity—emits light from two new families of trapped exciton states below the usual bright exciton: a directly emitting localized bright exciton about 40 meV lower and phonon-assisted side bands from localized momentum-dark excitons about 100–130 meV lower. The visibility of these features depends on temperature and on the width of the disorder potential, so the spectrum crosses from a localization-dominated regime below roughly 20 K to a phonon-dominated regime at higher temperatures. The authors build a microscopic theory in which the disorder potential quantizes the exciton center-of-mass motion, and they compute capture, escape, and intervalley scattering rates from exciton–phonon coupling. If correct, the results let experimenters read the width and depth of a trap from the photoluminescence fingerprint and design disorder potentials for single-photon emission.

What carries the argument

The central object is the localized-exciton basis obtained by projecting the photon-assisted polarization onto eigenstates of a single-particle eigenvalue problem for the exciton center-of-mass motion inside the Gaussian disorder potential, Eq. (1). The disorder potential of depth $V_0$ and width $\sigma$ quantizes the center-of-mass momentum into bound states $\chi^{\mu n}$; free-exciton wavefunctions $\phi^\mu$ come from the Wannier equation with a Keldysh potential. To keep free and localized states orthogonal, the continuum is described with orthogonalized plane waves, Eq. (S5), and all capture, escape, and intervalley scattering rates are computed from exciton–phonon matrix elements in the second-order Born–Markov approximation. The photoluminescence intensity is then an Elliott-type sum over localized bright states plus a phonon-assisted sum over localized dark states, so temperature enters through phonon occupations and the computed scattering rates. The machinery's job is to convert a single disorder parameter pair $(V_0,\sigma)$ into a full PL spectrum with peak positions, widths, and the temperature crossover.

What would settle it

Measure the photoluminescence of a WSe$_2$ monolayer with engineered Gaussian-like strain traps (e.g., nanopillars) at temperatures from 4 K to 100 K and track the intensity ratio of the localized bright peak ($\approx$40 meV below $X$) to the dark side bands ($\approx$100–130 meV below): the model predicts a crossover near 20 K, with $X_{\mathrm{Loc}}$ dominating below and dark side bands above, and no localized peaks when the trap width falls below the exciton Bohr radius ($\approx$1 nm).

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Extended reading notes

Core claim

The central claim is that in WSe$_2$ monolayers with a sufficiently broad Gaussian disorder potential, both bright $KK$ excitons and momentum-dark $K\Lambda$ and $KK'$ excitons form localized 1s states at energies below their free counterparts, and these localized states leave distinct photoluminescence resonances—$X_{\mathrm{Loc}}$ for the bright state and phonon-assisted $X^{D_{1,2}}_{\mathrm{Loc}}$ side bands for the dark states. The localized bright peak sits about 40 meV below the free exciton $X$, and the dark side bands appear roughly 100–130 meV below. The competition between phonon-driven capture into the trap and phonon-assisted intervalley scattering out of the bright state controls the intensities: below about 20 K capture wins and $X_{\mathrm{Loc}}$ dominates; above that temperature excitons scatter to dark valleys before capture and the dark localized side bands take over. The same mechanism explains a temperature-independent radiative linewidth of about 0.5–1.0 meV plus a linearly growing phonon contribution, matching the experimental 0.9 meV value, and it predicts that the peaks shift to lower energies and saturate as the disorder width grows. The authors emphasize that this yields two spectral regimes—disorder-dominated at low $T$, phonon-dominated at higher $T$—and that the features are most pronounced in tungsten-based TMDs because their dark excitons lie below the bright one.

Load-bearing premise

The calculation assumes that excitons entering any higher localized state relax almost instantly to the 1s state, so only 1s localized states matter for the optical response; if that timescale separation fails, the computed capture rates, peak intensities, and the 20 K crossover would change.

Editorial extensions

If this is right

Low-temperature PL of WSe$_2$ with strain or nanopillar traps should show a localized bright peak $X_{\mathrm{Loc}}$ about 40 meV below $X$ and phonon side bands from localized dark states about 100–130 meV below, as a fingerprint of trapping.
The crossover temperature near 20 K separates a localization-dominated regime (bright localized peak dominant) from a phonon-dominated regime (dark localized side bands dominant).
The radiative linewidth of localized excitons is about 0.5–1.0 meV and grows linearly with temperature from phonon scattering, reproducing the measured 0.9 meV.
For disorder widths below the exciton Bohr radius (about 1 nm) no trapping occurs and only free-exciton phonon side bands appear; as the width grows, the localized peaks redshift and saturate near 60 nm.
The intensity ratio between bright and dark localized resonances tracks the ratio of capture to intervalley scattering rates, so the spectrum can be used to infer trap characteristics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A temperature-dependent intensity ratio $I(X_{\mathrm{Loc}})/I(X^{D}_{\mathrm{Loc}})$ measured on a single sample with known trap width would directly test the predicted 20 K crossover—an experiment the paper does not report.
Applied to other TMDs, the model predicts that the visibility of dark localized side bands depends on the energetic ordering of the $K\Lambda$ and $KK'$ dark states, so molybdenum-based monolayers (where dark states sit higher) should show a different fingerprint.
The predicted capture-rate peaks whenever a new localized state enters the phonon window suggest that tuning $\sigma$ could resonantly enhance single-photon emission rates, a possible engineering lever.
The strong dependence on the disorder-width–depth product $V_0\sigma^2$ implies that deterministic strain patterning could be used to write arrays of localized emitters with tailored energies.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This paper extends the authors' earlier phonon-assisted PL framework to disorder-localized excitons in TMDs. The new piece is the simultaneous treatment of localized bright KK excitons and localized momentum-dark KΛ/KK' excitons. They predict that in WSe2, localized bright states produce a peak roughly 40 meV below the free exciton, and that dark localized states give phonon sidebands 100–130 meV below, with a temperature- and disorder-width-dependent crossover between localization- and phonon-dominated PL. That is a concrete, testable fingerprint and is genuinely new relative to their prior work.

What the paper does well: the formalism is standard and presented cleanly. They use DFT-parameterized tight-binding input, Keldysh screening, and compute capture and intervalley rates on the same footing. The predicted trends—peak shifts and saturation with increasing disorder width, linewidth growth with temperature—are physically sensible. The comparison to PLE data on binding energies is a useful anchor, and the qualitative agreement with nanopillar experiments is encouraging.

Now the soft spots, in proportion. The central predictions are not quantitatively compared with measured PL spectra. The well depth V0 is essentially set to reproduce the energy of the localized peaks in those experiments, so the position of XLoc is not an independent prediction. More importantly, the 1s-only truncation of the localized manifold is load-bearing. The authors assert that intra-localized relaxation is much faster than capture, but they never show the relevant rates. For V0 = 40 meV and σ = 30 nm the potential likely supports more than one bound state, and if capture lands first in an excited state that does not instantly relax, there should be additional bright peaks below X and the XLoc population—and hence the predicted crossover temperature—would change. The orthogonalized-plane-wave treatment of the continuum is a second uncontrolled ingredient in the same capture rates; no convergence check is provided. These are not fatal flaws—the qualitative regime picture may survive—but they make the paper's quantitative predictions (peak intensities, crossover at ~20 K) less solid than the text implies.

I'd send this to a serious referee. The gaps are fixable with a small calculation: show the intra-localized relaxation rates, check how many bound states are relevant, and ideally fit one measured spectrum at several temperatures. Even without that, the paper is a worthwhile contribution for people working on defect emitters in TMDs.

Referee Report

3 major / 6 minor

Summary. This manuscript develops a microscopic density-matrix and cluster-expansion model for excitons in monolayer WSe2 in the presence of Gaussian disorder potentials. Free excitons are described by the Wannier equation, and localized excitons are obtained by diagonalizing the center-of-mass Hamiltonian of Eq. (1), yielding bound states whose binding energies and wavefunctions depend on disorder width and depth. The authors then compute photoluminescence spectra from bright KK excitons, localized bright excitons XLoc, and phonon-assisted sidebands of momentum-dark KΛ and KK' excitons, with capture/escape rates obtained from second-order Born-Markov exciton-phonon scattering in an orthogonalized-plane-wave basis. The central predictions are: a localized bright peak roughly 40 meV below the free bright exciton at low temperature, phonon-assisted sidebands of localized dark excitons about 100-130 meV below the bright exciton, and a temperature- and disorder-width-dependent crossover between localization-dominated and phonon-dominated PL regimes.

Significance. If the central claims hold, the paper provides a unified microscopic interpretation of the many low-temperature PL resonances below the bright exciton in WSe2, connecting them to bright and momentum-dark localized excitons and to the competition between disorder capture and intervalley phonon scattering. The model's strengths are that it treats free and localized excitons on the same footing, computes capture rates and spectra from microscopic exciton-phonon matrix elements, and makes explicit, testable predictions about temperature and disorder-width trends, including a predicted T≈20 K crossover and radiative linewidths of 0.5-1.0 meV. The main limitations are that the absolute position of XLoc is essentially set by the input disorder depth, that quantitative comparison with measured PL spectra is not provided, and that two load-bearing approximations, the restriction to 1s localized states and the orthogonalized-plane-wave treatment of the continuum, are asserted without controlled numerical checks.

major comments (3)

[Main text, 'Photoluminescence of localized excitons', after Eq. (3), and Supplementary Material, Eq. (S6) discussion.] The restriction of the localized exciton manifold to the 1s state is load-bearing but is justified only by assertion. The main text states that 'intraexcitonic scattering of localized excitonic states (µn→µn′) appears on a much faster timescale than the capture processes itself,' and the Supplementary repeats that 'relaxation dynamics within the localized states happen on a much faster time scale than capture processes,' but no intra-localized relaxation rate is computed or compared with the capture and intervalley rates shown in the inset of Fig. 3. This matters directly for the central crossover claim: if excited localized states are not depopulated essentially instantaneously, they carry non-negligible occupation, produce additional bright PL peaks at smaller binding energies, and change the population of the 1s state that generates the predicted XLoc peak. The authors should provide a quantitative comparison of Γ(µn→µ1s) with Γcapture and Γintervalley for the parameters of Fig. 3, or include excited ns states explicitly and demonstrate that the spectra are converged.
[Supplementary Material, Eq. (S5), and main text discussion of capture rates.] The orthogonalized-plane-wave treatment of the continuum is an uncontrolled approximation that enters directly into the computed capture rates and hence into the predicted crossover. The normalization factor N_Q in Eq. (S5) contains the overlap sum ∑|⟨χµm_Q|φPW_Q⟩|², but the manuscript does not report how large this sum is for the parameters used, nor does it check the δ-function idealization φPW_Q ≈ δ_{Q,QF} against a numerically orthogonalized continuum basis. Since the capture rates in the inset of Fig. 3 and the consequent visibility of XLoc versus dark sidebands depend on these overlaps, I request a convergence or consistency check, for example by comparing with continuum states obtained from the Wannier equation orthogonalized to the localized states, and by reporting the magnitude of the projection corrections.
[Main text, results 'Photoluminescence of localized excitons' and 'Disorder-induced control of PL'.] The claim that 'calculated optical spectra agree well with recent experiments' is not quantitatively supported, and the position of the main XLoc peak is not an independent prediction. In Fig. 3 the XLoc peak appears approximately 40 meV below X because V0 = 40 meV was chosen from the nanopillar experiments of Ref. [21], so the peak position essentially follows from the input disorder depth. The more robust predictions are the relative intensity trends with temperature and disorder width, the linewidth behavior, and the dark-localized sidebands. I recommend that the authors explicitly separate input-determined features from emergent model predictions and, where possible, overlay calculated PL spectra on measured spectra (e.g., the data of Refs. [21, 58, 59]) to substantiate the claimed agreement.

minor comments (6)

[Theoretical approach.] There is a typo 'resultign' in the sentence introducing the photon-assisted polarization; it should be 'resulting'.
[Results, temperature-dependent PL.] The text 'phonon side[17, 32]bands' contains misplaced citation brackets; the phrase should read 'phonon sidebands' with citations following.
[Disorder-induced control of PL.] In the paragraph on narrow disorder potentials, 'we are in the the phonon-dominated regime' contains a duplicated article and should be corrected.
[Eq. (1).] The eigenvalue equation as printed contains 'εµnχµxn Q' on the right-hand side, which appears to be a typo for 'εµnχµn Q'; please check the equation.
[Supplementary Material, Eq. (S5).] The notation ⟨χµm_Q|φPW_Q⟩ is ambiguous because the localized states carry the same Q index as the plane-wave state; please clarify the labeling of the basis states in the orthogonalization sum.
[Figures 2 and 3.] The paper uses V0 = 120 meV for Fig. 2 but V0 = 40 meV for Figs. 3 and 4; a short remark explaining how the fixed-depth curves of Fig. 2 relate to the lower-depth spectra of Figs. 3-4 would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: localized-state energies and capture/intervalley rates are derived from the model parameters, not fitted to the predicted PL features.

full rationale

The paper's central predictions are derived rather than assumed. The XLoc peak position near 40 meV below X is obtained by solving the localized-exciton eigenvalue problem, Eq. (1), with a Gaussian disorder potential of depth V0 = 40 meV chosen from independent experimental reports on nanopillars and local strain ([21]); the eigenenergy is a computed function of V0 and width, not an identity or a fit to the PL spectrum. The temperature-dependent crossover between localization- and phonon-dominated regimes is controlled by microscopically computed capture and intervalley exciton-phonon scattering rates (Eqs. S3-S7), none of which are adjusted to reproduce the predicted spectra. The paper does cite previous work by the same authors ([17], [32], [63], [64]) for the underlying TMD Bloch equations and exciton-phonon matrix elements, but those citations supply the standard many-body formalism and are not invoked as external proof of the target result, so they are not circular. The 1s-only truncation of localized states, justified by an asserted timescale separation in the Supplementary, is an unvalidated approximation and a potential correctness risk, but it is not a circular step: it does not define the predicted quantities in terms of themselves. No step in the derivation reduces, by construction or by fitted parameter renaming, to its own input.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

No new physical entities are introduced; localized excitons are bound states of the model with a Gaussian potential. The free parameters V0 and sigma are set by comparison with specific experiments, and the central approximations are the single-1s-state reduction and the orthogonalized-plane-wave treatment of the continuum.

free parameters (2)

disorder depth V0 = 120 meV (Fig. 2), 40 meV (Figs. 3-4)
Chosen to match PLE measurements [27] and nanopillar experiments [21], not derived from the model.
disorder width sigma = varied 1-60 nm; fixed 30 nm for temperature study
Swept to study localization; 30 nm chosen to match nanopillar/strain experiments [21].

assumptions (6)

standard math Wannier equation with Keldysh potential gives free-exciton wavefunctions and energies
Standard semiconductor model for excitons in 2D; used throughout, e.g., Eq. (1).
domain assumption Cluster expansion / density matrix formalism truncation at the excitonic two-particle level
Assumes higher-order correlations are negligible; standard in this field but an approximation.
domain assumption Second-order Born-Markov approximation for exciton-phonon scattering and capture rates
Used in supplementary Eq. (S3) to derive rates; assumes Markovian and weak coupling.
ad hoc to paper Orthogonalized plane waves for free continuum states
Introduced to maintain orthogonality between localized and free states (supplementary Eq. S5); not a standard approach and could affect rates.
ad hoc to paper Only the 1s localized state contributes to PL
Justified by fast intra-localized relaxation, asserted but not numerically verified.
domain assumption Phonons are in a Bose equilibrium bath
Bath approximation used for phonon occupations eta; standard.

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Cite this review

Pith. "Pith review of Optical fingerprint of bright and dark localized excitonic states in atomically thin 2D materials." pith.science (2026). https://pith.science/paper/IRHZDZ7Z

@misc{pith2026190805071,
  author       = {Pith},
  title        = {Pith review of: Optical fingerprint of bright and dark localized excitonic states in atomically thin 2D materials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IRHZDZ7Z}},
  note         = {Machine review of arXiv:1908.05071}
}

read the original abstract

Point defects, local strain or impurities can crucially impact the optical response of atomically thin two-dimensional materials as they offer trapping potentials for excitons. These trapped excitons appear in photoluminescence spectra as new resonances below the bright exciton that can even be exploited for single photon emission. While large progress has been made in deterministically introducing defects, only little is known about their impact on the optical fingerprint of 2D materials. Here, based on a microscopic approach we reveal direct signatures of localized bright excitonic states as well as indirect phonon-assisted side bands of localized momentum-dark excitons. The visibility of localized excitons strongly depends on temperature and disorder potential width. This results in different regimes, where either the bright or dark localized states are dominant in optical spectra. We trace back this behavior to an interplay between disorder-induced exciton capture and intervalley exciton-phonon scattering processes.

Figures

Figures reproduced from arXiv: 1908.05071 by the authors.

**Figure 1.** We model both free and localized excitons on the same microscopic footing. While free excitons are formed by a Coulomb potential leading to the quantization of the relative motion of electron and hole, localized excitons are formed due to trapping into a disorder potential giving rise to a quantization of the exciton center-of-mass motion. We show that depending on the width of the trapping potential, the existenc… view at source ↗

**Figure 2.** [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Temperature dependent photoluminescence. (a) Surface plot of time integrated photoluminescence for different temperatures and a fixed disorder potential with σ = 30 nm and V0 = 40 meV. (b) 2D cuts from the surface plot at constant temperatures. Note that the spectra are shifted in energy so that the bright X exciton is located at 0 meV. We observe a variety of new disorder-induced peaks below the bright exciton inc… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Disorder-width dependent photoluminescence. (a) Time-integrated photoluminescence spectra at 10 K for varying disorder width. (b) 2D cuts of the surface plot at four fixed widths. If the disorder potential is too narrow (1 nm, blue line), excitons can not be trapped a…

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