Unveiling the True Lifetime Distribution of Silicon Nanocrystals: A Combined Time‑Resolved and Frequency‑Resolved Spectroscopic Study

Abstract

Silicon nanocrystals (SiNCs) exhibit complex luminescence dynamics that arise from size‑dependent radiative and non‑radiative rates, inhomogeneous broadening, and inter‑particle interactions. These factors generate non‑exponential decay profiles, typically modeled by a stretched exponential (SE) function. In this work, we first derive the population decay function for a decay of the form “exp[−(t/τ)^{β}]”, and we compare the lifetime distributions obtained when the luminescence decay or the population decay follows this function. For β values well below 1, the two approaches yield markedly different mean lifetimes. We apply both SE models, a lognormal distribution, and a bimolecular decay to luminescence data from two thermally grown SiNC ensembles with distinct mean sizes. The mean lifetimes are strongly dependent on the experimental configuration and the chosen fitting model, and none of the conventional models adequately captures the ensemble decay dynamics. We then employ frequency‑resolved spectroscopy (FRS) to extract the lifetime distribution directly, revealing a half‑width of ~0.5 decades and a distribution resembling a high‑frequency‑skewed lognormal function. The combined use of time‑resolved and frequency‑resolved techniques provides the most reliable insight into the luminescence dynamics of nanocrystals with broad emission spectra.

Introduction

Colloidal nanoparticles are integral to catalysis, biomedicine, and optoelectronics [1–4]. Semiconductor nanocrystals (NCs) are especially attractive for light emission, photovoltaics, and photocatalysis [5,6]. Silicon nanocrystals (SiNCs) have attracted attention due to their tunable emission, abundant and biocompatible silicon core, and potential for scalable synthesis [7–9]. Accurate knowledge of their optoelectronic properties is essential for device development, and time‑resolved spectroscopy is a primary tool for this purpose.

SiNC luminescence lifetimes are commonly modeled by a stretched exponential (SE) function, exp[−(λt)^{β}]”, where 0 < β ≤ 1 is the dispersion parameter, λ is the decay rate, and t is time. This form implies an asymmetric distribution of decay rates that tail toward longer lifetimes. Once β and λ are obtained from fitting a decay curve, the underlying rate distribution can be approximated [9].

The physical origin of SE decay in silicon and other quantum dots has been debated for two decades, with proposed explanations including carrier tunneling and trapping in densely packed ensembles [11], size‑dependent electron‑phonon coupling [10], and a distribution of non‑radiative barrier heights [13,14]. A clear understanding of the rate distribution is thus essential for elucidating the luminescence mechanism in SiNCs and related systems.

Many prior studies assume a priori the SE model, often neglecting alternative distributions and the detector’s responsivity over the broad SiNC emission spectrum. This oversight hampers comparability across studies and obscures the true decay dynamics. Furthermore, frequency‑resolved spectroscopy (FRS) has not been applied to SiNCs, although it offers a model‑free route to lifetime distributions.

The goal of this paper is to establish a rigorous framework for measuring, modeling, and interpreting SiNC luminescence dynamics. We aim to resolve inconsistencies in the literature, provide a more reliable comparison of different datasets, and deepen the understanding of the underlying mechanisms.

Basic Theory

We compare three decay models: the stretched exponential (SE), a lognormal distribution, and a bimolecular decay. For any model, the emission probability density function g(t) is related to the fraction of excitations remaining at time t by

∫₀ᵗ g(t′) dt = 1 − c_t/c₀,

where c_t and c₀ are the excited NC counts at time t and initially. For a first‑order (monomolecular) recombination, dc_t/dt = −λc_t, leading to c_t/c₀ = exp(−λt) and g(t) = λ exp(−λt). Including both radiative (λ_R) and non‑radiative (λ_NR) rates, the measurable intensity becomes g(t) = λ_R exp[−(λ_R+λ_NR)t].

When the population decays as a stretched exponential, c_t/c₀ = exp[−(λ_SE t)^{β}], the emission probability function is

g(t) = β λ_SE^{β} t^{β−1} exp[−(λ_SE t)^{β}].

Using inverse Laplace transforms, the corresponding rate distribution H(λ) widens and skews toward high frequencies as β decreases. However, this form cannot be decomposed into separate radiative and non‑radiative contributions, and it is strictly normalized only when λ_NR = 0.

For the intensity decay model g(t) = (λ_SE β/Γ(1/β)) exp[−(λ_SE t)^{β}], the population decay is obtained via

c_t/c₀ = 1/Γ(1/β) Γ[1/β, (λ_SE t)^{β}],

leading to mean lifetimes

⟨τ_SE⟩ = τ_SE Γ(2/β)/Γ(1/β), ⟨t⟩ = τ_SE Γ(3/β)/(2Γ(2/β)).

The lognormal decay rate distribution is defined as

H(λ) = (1/λ) (1/(σ√{2π})) exp[−(ln λ − μ)²/(2σ²)],

with median rate exp(μ), mean exp(μ+σ²/2), and most probable lifetime exp(μ−σ²). The resulting intensity is the Laplace transform of H(λ):

I_t = A ∫₀^∞ H(λ) exp(−λt) dλ + dc.

The bimolecular decay follows dc_t/dt = −λc_t², yielding c_t/c₀ = 1/(λc₀t+1) and a power‑law intensity I_t/I₀ = A λc₀/(λc₀+1)². This model has no finite mean lifetime.

Figure 1 illustrates the differences between the two SE formulations and their corresponding rate distributions.

Quadrature Frequency‑Resolved Spectroscopy

Quadrature FRS (QFRS) employs a sinusoidally modulated pump beam at angular frequency ω and records the phase and amplitude of the photoluminescence (PL) response. The quadrature component Q_PL = Z_PL sin(Δθ_PL) provides a direct measure of the lifetime distribution after normalizing by the laser modulation amplitude Z_LA.

The QFRS signal for a single exponential decay is given by

S_{log10 r} = (ω τ₀)/(1+ω² τ₀²),

with τ₀ = 1/ω₀. Since a realistic decay distribution spans several decades, a deconvolution is required to extract the true distribution.

Results and Discussion

Basic Characterization

Bright‑field TEM images were used to estimate particle diameters by pixel counting. The resulting size distributions were well described by lognormal functions with mean diameters of 2.9 nm (σ=0.1555) for the 1100 °C annealed sample and 5.4 nm (σ=0.1917) for the 1200 °C sample. High‑resolution TEM confirmed the lattice fringes and size estimates. FTIR and XPS verified successful dodecene functionalization; the smaller particles exhibited greater oxidation and lower functionalization density.

Photoluminescence and Time‑Resolved Spectroscopy

The PL spectra peaked at 660 nm (FWHM = 123 nm) for small NCs and 825 nm (FWHM = 198 nm) for large NCs. Using the size‑dependent bandgap formula E_g = √{E_{g,bulk}² + D/R²} with D = 4.8 eV²/nm², we obtained predicted bandgaps of 1.87 eV (small) and 1.37 eV (large), consistent with the measured PL peaks. Absolute quantum yields (AQY) were 12 % (small) and 56 % (large); independent measurements yielded 18 % and 48 %, respectively, within the typical 10 % uncertainty range.

All samples exhibited non‑exponential decay profiles. Fits using Eqs. 5, 9, 16, and 17 (SE, lognormal, bimolecular) were performed via least‑squares minimization. Residuals oscillated for all models, indicating incomplete capture of the decay shape; the simple SE (Eq. 9) and lognormal (Eq. 16) provided the lowest sum‑of‑squares values. Mean lifetimes varied strongly with the chosen model (Table 2). The Higashi‑Kastner method, which locates the peak of I_t·t versus t, yielded decay constants similar to (1/β)^{1/β}·τ_SE derived from Eq. 9. The bimolecular model poorly described the data, consistent with isolated NCs not experiencing high exciton densities.

Excitation conditions were estimated using absorption cross‑sections (~10⁻¹⁴ cm²) and an irradiance of 4500 W m⁻² at 352 nm. The average excitations per NC were < 1 for large NCs and ≈ 0.2 for small NCs, indicating that over‑excitation effects were negligible, as confirmed by power‑dependent lifetime measurements showing < 2 % variation.

To probe wavelength dependence, PL decays were measured at discrete emission wavelengths (3 nm bandpass) using a monochromator. The dispersion parameter β approached unity for longer wavelengths, and lifetimes increased with wavelength (Fig. 5, Table 3). Shorter particles consistently exhibited shorter lifetimes than larger ones at the same wavelength, reflecting lower AQY and higher non‑radiative surface contributions.

Frequency‑Resolved Spectroscopy

FRS was validated using a known RC circuit (mono‑exponential decay, τ = 78.9 µs) and a Eu‑chelate‑doped microsphere (τ ≈ 670 µs). The QFRS spectra matched the expected response function (Eq. 18) with peak frequencies of 12.7 kHz and 1570 Hz, respectively.

SiNC FRS data were only slightly broader than the intrinsic response, necessitating deconvolution via the Richardson‑Lucy algorithm to retrieve the lifetime distribution. The deconvolved distributions (Fig. 6) spanned ~0.5 decades, with the small NCs showing a symmetric log‑scale shape and a peak at 19.9 kHz (50.3 µs), and the large NCs peaking at 6.28 kHz (159.2 µs). SE and lognormal model fits yielded different distribution shapes and peak frequencies, all lower than the QFRS peaks, reflecting the sensitivity of steady‑state TRS to pulse duration and detector responsivity.

Detector responsivity significantly affected TRS results; APD and PMT measurements differed by a factor of ~2 in mean decay times. The APD peaked at 600 nm, while the PMT peaked at 850 nm, leading to divergent “best” model fits. Correcting for responsivity in FRS is feasible if the spectral dependence of the decay rate distribution is known; TRS lacks a straightforward correction method.

Conclusions

We derived the population decay corresponding to the commonly used SE luminescence decay and provided expressions for mean lifetimes. Two thermally grown, dodecene‑functionalized SiNC samples (mean diameters 2.9 nm and 5.4 nm) were synthesized and characterized. TRS data fitted with SE, lognormal, and bimolecular models failed to fully capture the decay shape; mean lifetimes depended on detector responsivity and model choice. Frequency‑resolved spectroscopy, after deconvolution, yielded the lifetime distribution directly, revealing a ~0.5‑decade spread and a high‑frequency skew for large NCs. FRS is therefore the preferred method for probing SiNC luminescence dynamics, though careful deconvolution and detector calibration are essential. Future studies should incorporate detector responsivity corrections and avoid defaulting to SE models without validating the underlying distribution.

Methods

SiNCs were prepared by annealing 4 g of hydrogen silsesquioxane (HSQ) at 1100 °C or 1200 °C for 1 h in 5 % H₂/95 % Ar. The resulting SiNC/SiO₂ composites were ground, HF‑etched to remove the silica matrix, and functionalized with 1‑dodecene (≈ 4.6 mmol) and AIBN (20 mg) under Ar. After 24 h at 70 °C, the reaction mixture was precipitated with methanol/ethanol, centrifuged, and re‑dispersed in toluene. The final dodecyl‑terminated SiNCs were stored in 5 mL dry toluene (≈ 0.5 mg mL⁻¹).

TEM samples were prepared on ultrathin (≈ 3 nm) carbon‑coated copper grids and imaged with a JEOL JEM‑2010 (bright‑field) and JEOL JEM‑ARM200CF (HRTEM). FTIR spectra were recorded on a Nicolet 8700, and XPS was performed on a SPECS system (Mg Kα source).

PL excitation used a 352‑nm Ar⁺ laser pulsed (50 % duty cycle, 50–250 Hz) via an acousto‑optic modulator (AOM). The beam was split by a beamsplitter; the main portion illuminated the sample cuvette, and the emitted PL was collected by a 0.22 NA fiber, filtered by a 450‑nm long‑pass filter, and directed to the detector. The PL spectrum was recorded on an Ocean Optics miniature spectrometer, calibrated with a HL‑3 + -CAL source. Quantum yield was measured using an integrating sphere (405‑nm excitation) on a diluted solution (A ≈ 0.15).

Time‑resolved PL was recorded with either a Thorlabs 120A2 avalanche photodiode (APD) or a Hamamatsu h7422‑50 photomultiplier tube (PMT), interfaced to a Moku:Lab oscilloscope or a Becker‑Hickl PMS400 multiscaler. Detector response functions were measured and applied during analysis. QFRS experiments employed a sinusoidally modulated AOM; the reference signal was derived from a Thorlabs PDA10A photodiode. The PL response was collected by the APD and processed with a lock‑in amplifier to extract in‑phase and quadrature components.

For fast decay component detection, a 405‑nm picosecond diode laser (Alphalas GmbH) excited the NCs while a Becker‑Hickl HPM‑100‑50 PMT (100 ps response) and an SPC‑130 pulse counter recorded the PL. No nanosecond decay was observed.

Abbreviations

APD

Avalanche photodiode

AQY

Absolute quantum yield

FRS

Frequency‑resolved spectroscopy

LN

Lognormal

NCs

Nanocrystals

PL

Photoluminescence

PMT

Photomultiplier tube

QFRS

Quadrature frequency‑resolved spectroscopy

SE

Stretched exponential

SiNCs

Silicon nanocrystals

TRS

Time‑resolved spectroscopy