Origin of stretched-exponential photoluminescence relaxation in size-separated silicon nanocrystals

Silicon nanocrystals (SiNCs) offer a potential green alternative to metal-chalcogenide nanocrystals because of the non-toxicity,¹ earth abundance, and technological familiarity of silicon. With opportunities for applications in photovoltaics,^2–5 LEDs,^6–9 and biological sensors,^10–12 a thorough understanding of exciton recombination mechanisms in SiNCs will be important for optimizing their potential in future technologies. However, silicon has an indirect bandgap, which requires a finite momentum shift for radiative recombination. To complicate things further, additional much faster radiative relaxation timescales have also been observed.^13–16 In our recent work, we demonstrated that hydrocarbon-passivated SiNCs can exhibit both slow (μs) and fast (ns) PL relaxation, where both modes appear to impact the overall quantum yield (QY).¹⁶ Similarly, other groups have demonstrated that with appropriate passivation, surface related relaxation channels give rise to fast relaxation modes with nanosecond lifetimes.^14,17,18 It is now commonly accepted that these surface effects can have considerable influence on the overall luminescent properties, leading to several interesting topics of research.¹⁹ For example, Dasog and coworkers were able to exploit different passivation schemes and achieve ligand dependent emission across the entire visible spectrum.¹⁷ In another recent study, Li et. al., showed that surface related relaxation channels can be highly emissive, reaching QYs greater than 90 % while maintaining a relatively narrow FWHM.¹⁸ Although several interesting and potentially fruitful opportunities related to the surface passivation of SiNCs exist, a complete understanding of both the surface and core emission is necessary to further optimize the recombination dynamics of these potentially useful materials.

Assigning specific emission pathways to either the SiNC core or surface has not been without controversy, yet the most modern pictures point to the long microsecond decay as arising from the core.^{13–15,17,20–23} However, even within this widely accepted view, the stretched nature of the decay has not been fully reconciled or explained. Reported values for the stretching exponent vary from 0.4 to 1, with precise explanations invoking size polydispersity or electron-phonon coupling.^24–26 These challenges primarily arise from the indirect band structure of silicon, the sensitivity of the optical and electrical properties to surface passivation, and challenges associated with synthesizing and preparing monodisperse samples. Synthetic methods for SiNCs^27–31 are more complex or hazardous than the hot injection methods allowed by the metal-chalcogenide nanocrystals.³² This has led to the development of post-synthetic size-separation methods, which are often necessary for achieving well-resolved measurements of size dependent SiNC properties.^33,34

In this work, we analyze the stretched microsecond PL decay of size-purified SiNCs in both the solvated and solvent-free state by varying the spectral window of the measured response. Typically, the slow, microsecond radiative relaxation is modeled with the stretched exponential decay function, $I(t)=I0exp−tτstrα$⁠. Here, we use the specific model formulated by Johnston to map the stretched exponential decay onto a continuous sum of exponential relaxations and the appropriate PDF of PL relaxation rates.³⁵ An interpretation of the stretched decay dynamics through a PDF and a continuous sum of exponential relaxations has been done previously,^25,36 as has the use of spectrally resolved PL lifetime measurements to study the radiative recombination kinetics of SiNCs.²⁶ However, to the best of our knowledge, this is the first time that the two approaches have been used in combination to analyze the radiative recombination dynamics of monodisperse, size-purified SiNCs. By doing so, we gain additional new insight into the exciton-phonon coupling that underlies PL relaxation in quantum-confined silicon.³⁷ 

The SiNCs used here were chemically synthesized in our lab by the lead author using solution-state methods described elsewhere.³⁴ They were surface passivated with 1-decene and then dispersed in hexane.³⁴ This produces nanocrystals that are approximately 3 to 4 nm in diameter, exhibiting red PL (inset, Fig.1b). The as-produced (AP) SiNC solution had a solvent dependent QY of approximately 20 % in hexane, and although the colloidal SiNCs were stored in an inert atmosphere, a drop in QY, a slight blue shift in the PL, and the emergence of a weak blue emission component occurred over time, which we attribute to age-related processes such as oxidation (Fig. S2 of the supplementary material). Aliquots of the parent sample were dispersed in various solvents under an inert nitrogen atmosphere to limit oxidation. The dried samples were then dispersed in similar volumes of various solvents - such as THF, toluene, and hexane - and the subsequent QY measured (Fig. 1a). Hexane showed the highest QY, followed by toluene and then THF, which showed a considerable drop in the overall QY. One possible explanation for the observed drop in PL efficiency is surface oxidation resulting from the solvent itself. However, if this were the case one would expect that the drop in QY would be irreversible. Yet, when one aliquot was 1) dispersed in hexane, 2) dried under inert conditions, 3) dispersed in THF, 4) dried again under inert conditions, and 5) re-dispersed in hexane, a degree of reversibility in the QY was observed (inset, Fig. 1a). Although the sample did undergo a degree of irreversible degradation, there was an increase in the final QY relative to the QY in THF, indicating that this quenching was reversible and surface oxidation was not the sole cause for the QY drop. Therefore, the solvent itself must play a role in the observed QY.

To further probe this, we measured the QY of the parent material in several more solvents. Plotting the QY as a function of solvent donor number (Fig. 2b) suggests that a larger donor number corresponds to a lower QY. A similar trend between QY and the dielectric constant of the solvent was also observed (Fig. S1 of the supplementary material).

Recent experimental work by Wheeler et. al., suggests that hypervalent bonding of the solvent to chlorine-passivated plasma-synthesized SiNCs effectively induces colloidal stability without the need for a hydrocarbon passivating layer.³⁸ Similarly, Shu et. al. explained the dramatic increase in the PL of plasma-synthesized SiNCs by considering the hypervalent interaction of silyl groups with the nanocrystal surface.³⁹ By heating a solution of SiNCs to relatively low temperatures (< 200 ^oC), they detected the desorption of such moieties from the nanocrystal surface accompanied by a substantial increase in PL. Conical intersections related to changes to the potential energy surface due to hypervalent interactions were then invoked as a way to explain the substantial decrease in the observed PL when silyl moieties are present.³⁹ 

Furthermore, theoretical work by the Levine group has shed light on the impact that different defects have on the optical properties of SiNCs,^39–41 with the finding that oxidation in silicon nanocrystals can open conical intersections that allow for an efficient non-radiative recombination channel. In the QY vs. solvent data presented here, a similar argument can be made. One could imagine that certain solvents would interact with the SiNC surface through hypervalent bonding. This interaction alters the native potential energy surface, resulting in a non-radiative channel that subsequently reduces the overall QY. A solvent with a higher donor number would interact more strongly with the SiNC surface, causing a greater number of non-radiative pathways, which would in turn lead to a greater decrease in QY (Fig. 1b). Another possible explanation invokes dielectric stabilization of the localized wavefunction at surface oxidation sites. As predicted by the Levine group, higher dielectric constant solvents stabilize the localized wavefunction near such a defect, increasing the likelihood of recombination through this type of channel.⁴¹ In the simplest models, the QY can be related to the non-radiative (k_nr) and radiative (k_r) recombination rates as QY = k_r/(k_r + k_nr) with τ = (k_r + k_nr)⁻¹.⁴² Higher dielectric constant solvents would thus increase k_nr and lower the overall QY.

Turning our attention to the precise mechanism of radiative relaxation in SiNCs, monodisperse ‘super-fractions’ were prepared by separating the AP material into size-resolved fractions using density gradient ultracentrifugation (DGU). More information on the separation process can be found in the supplementary material. The DGU purification process was done nine separate times and individual fractions with near identical PL spectra were combined to form ‘super’ fractions. Each such fraction was labeled as SA, SB, SC for super-fraction A, super-fraction B, super-fraction C, ect. The PL of each super fraction can be seen in Fig. 1d, where the wavelength of peak emission increases with size, typical of a quantum confined system. We used hexane, the solvent that induces the least amount of non-radiative relaxation, and measured the QY of each fraction. The inset to Fig. 1c shows the QY as a function of the peak emission wavelength, while QY vs. size is plotted in Fig. 1c, where the conversion from peak emission energy to nanocrystal size R was executed through $E=Eo+3.732R1.39$ with E_o = 1.17 eV.⁴³ We note that the highest QY of the fractions (40 %) is well above that of the parent suspension. The QY quickly diminishes as the size of the nanocrystal decreases, indicating an increasing number of non-radiative relaxation channels.

The time dependent PL relaxation kinetics of each size-resolved SiNC suspension (excluding SA and SB for which the QY was deemed too low) were measured in two different states (solvent-free and solvated) and the data were fitted with a stretched exponential relaxation; I_cont(t) = A exp[−(t/τ_str)^α]. In the solution state, the measured decays were spectrally resolved, i.e. a monochromator was used to allow only the peak spectral emission from the samples to reach the detector (5 nm resolution). Little difference was observed between the solution and solvent-free decays. However, when examining the spectrally resolved peak emission, the lifetime and stretching exponent both substantially increased, with α approaching 1 (Fig. 2a). The parameters τ_str and α are shown as a function of peak emission wavelength in Figs. 2c–d.

To facilitate numerical interpolation, the data in Figs. 2c–d were fit with a quadratic function of the form y = A + Bx + Cx², where the resulting coefficients are listed in Table S1 of the supplementary material. Because there was a negligible difference between the response of the solution vs. solvent-free but a strong dependence on the detection scheme (FS vs. SR), the FS data were fit to a quadratic function that included both solution and solvent-free data, while the SR data was fit to a separate and distinct quadratic function. The quadratic fitting schemes have no physical significance but are used in the analysis to interpolate lifetime and stretching exponent as a function of peak emission wavelength.

Stretched exponential functions are typically used to model a system that contains a distribution of decay rates and lifetimes.⁴⁴ Hence, the stretched exponential relaxation function can be written as a continuous sum of distinct exponential decays weighted by the appropriate PDF;

where P(s, α) is the PDF, $s=τstrτ$ is a dimensionless rate, and I_cont(t) is the total intensity at time t. The normalization of the PDF is chosen such that $∫0∞Ps,αds=1$⁠. The PDF can be expressed as³⁵ 

Johnston³⁵ gives significantly more detail on the origin of Eq. 2 and the curious reader is directed to Ref. 35 for more details. For computational purposes, Eq. 1 can be approximated as a discrete sum of exponential decays through

Adjusting the upper and lower limit, UL and LL, respectively, effectively acts as a lifetime ‘filter’ to map the decays in a spectrally resolved measurement to their corresponding location in the PDF. Thus, Eq. 3 can be written as

For the FS configuration, PDFs were generated from Eq. 2 using Mathematica® and are plotted in Fig. 3a. To demonstrate the equivalence of Eq. 1 and Eq. 4, Mathematica® was again used to generate stretched exponential decays from Eq. 4, where the PDF used was on either side of the experimentally relevant window; α_FS = 0.60 and α_FS = 0.80. LL = 0.1, UL = 20 and Δs = 0.1 were used to achieve a reasonable approximation to Eq. 1, as shown in Fig. 3b. Johnston noted that the UL is related to τ_FS, and even though the function was summed to UL = 20 (significantly past the relevant decays), the fact that the summation is not infinite and the discrete nature of Eq. 4 lead to the observed discrepancy between I_cont(t) and I_disc(t).

Simply changing UL and LL in Eq. 4 can drastically alter the resulting decay. Figures 4a–b show the result from Eq. 4 with the LL and UL set to 0.1 and 20, respectively, for α_FS = 0.60 (Fig. 4a) and α_FS = 0.80 (Fig. 4c) indicated in black. The colored background decays are matched with the corresponding region of the PDF found in Figs. 4b & e. For example, a lower limit of s = 0.01 and an upper limit of s = 0.10 in Fig. 4b is blue and corresponds to the longest blue decay in Fig. 4a, while in a similar fashion setting the limits to LL = 1.91 and UL = 2.00 corresponds to the yellow part of the PDF and the corresponding decay. By changing the upper and lower limits in the sum, one can effectively modify the resulting decay curve. Furthermore, P(s, α_FS) was converted to P(τ, α_FS) by noting that $s=τstrτ$ and then interpolating τ_str from Fig. 2c. These are again color coded so that the appropriate limits match the resulting curve, I_discSR(t).

The stretched lifetime, τ_str, is commonly assumed to be either the most probable lifetime, but it becomes apparent that this is not the case, as the most probable lifetime (the peak of the PDF) significantly differs from τ_str (black dotted lines in Fig. 4). There is almost an order of magnitude difference between the most probable decay time and τ_str. The distribution of decay times matches intuition, however. As α → 1, the width of the PDF also decreases and approaches a delta function.

We then set out to fit the spectrally resolved data using Eq. 4 by only modifying the upper and lower limit:

The results are shown in Fig. 5, where the full spectrum PDF is plotted as a function of τ for α_FS ranging from 0.60 to 0.80 in steps of 0.05. The vertical dashed blue and green line represents $τstrFS$ and UL in Eq. 5, respectively (note that s and τ are the inverse of each other). The FS stretched lifetime increases in accordance with Fig. 2c while UL similarly increases as a function of peak emission (Fig. 6b). Interestingly, as the size of the nanocrystal increases so does the lower limit of lifetimes used to fit the spectrally resolved data (Fig. 6), indicating that a small subset of the total decays make up the majority the observed PL. At the same time, α_FS is also increasing and thus the PDF is becoming narrower towards the upper end of the spectral window examined in this study. As mentioned earlier, a simple model relating the QY to lifetime is QY = k_r/(k_r + k_nr) with τ = (k_r + k_nr)⁻¹. Typically, τ_str is compared to the QY to demonstrate this relationship, but we can see that τ_str does not represent the most meaningful decay time. Rather, a more appropriate value is the most probable decay time, which is shown in Fig. 7a. The most probable decay time was thus found from the PDFs of Fig. 5 and is plotted next to the solvated QYs of the fractions in Fig. 7b, demonstrating reasonable qualitative agreement.

For further analysis of the stretched nature of PL relaxation, linear variable filters were used to select a ‘blue’, ‘peak’, and ‘red’ region of the PL spectrum and the subsequent spectrally resolved PL relaxation was measured. The results are shown in Fig. 8 for fraction SF. From this data, we see that the mapping of PL to PDF is such that higher energy photons correspond to shorter decay times and lower QY, while lower energy photons correlate with longer decay times and higher QY. Furthermore, the emission that is spectrally weighted toward the ‘peak’ and the ‘red’ end of the PL spectrum exhibits a stretching exponent closer to 1, while that closer to the ‘blue’ end of the spectrum is more stretched. Because the SiNCs used here are relatively monodisperse, we can ascribe the majority of the PL spectral line-width to electron-phonon coupling and the indirect nature of the bandgap. In this view, the results in Fig. 8 are telling and can be qualitatively interpreted within the context of a simple model of electron-phonon coupling.¹⁶ For a given SiNC fraction or nanocrystal size, the ‘redder’ emission is more radiatively efficient and corresponds to a transition from the bottom of the band edge and the subsequent emission of a phonon to satisfy momentum conservation. In contrast, the ‘bluer’ emission is less radiatively efficient and corresponds to stronger phonon coupling from a slightly higher energy state associated with higher-order structure just above the band edge and the absorption of a phonon or even multiple phonons. In this simple picture, as nanocrystal size decreases, the latter effect would become more prominent.

Using size-resolved SiNC fractions dispersed in the optimal solvent for efficient radiative recombination, we have examined the role of nanocrystal size and spectral resolution in the slow, stretched-exponential PL relaxation of colloidal SiNCs. Based on the stretching exponents deduced from the measured PL decay, and using a concise model of stretched-exponential relaxation available in the literature, we were able to extract the PDF for radiative decay time as a function of both nanocrystal size and the energy of the emitted photons. Our results suggest a correlation between larger nanocrystals, ‘stretching’ exponents closer to unity, longer PL lifetime, and more efficient emission. For a given nanocrystal size, our results also suggest that the spectral portion of the PL that resides both near and on the ‘red’ or lower-energy side of the emission peak corresponds to a lesser degree of ‘stretching’, longer PL lifetime, and higher emission efficiency. In contrast, the emission on the ‘bluer’ or higher-energy side of the PL peak correlates with shorter decay time, a greater degree of stretching, and lower overall emission efficiency. Because the fractions are monodisperse, we suggest that the underlying mechanism relates predominantly to the strength and nature of electron-phonon coupling. Our results help further clarify the multi-exponential nature of PL relaxation in nanocrystalline silicon while potentially providing additional insight into how the quantum yield might be further optimized.

SiNC were synthesized from solution using methods described elsewhere.³⁴ Briefly, silicon tetrachloride was exposed to a controlled amount of water under a nitrogen atmosphere to create a silicon suboxide. This was then annealed in a tube furnace under a slightly reducing atmosphere of 5 % N₂ and 95 % Ar. Under these conditions, the silicon suboxide decomposes into SiNCs encased in a silicon dioxide layer. A mortar and pestle were used to mechanically breakdown the composite and then HF acid was used to liberate the nanoparticles from the silicon dioxide layer. The bare nanocrystals were transported to a round bottom flask connected to a Schlenk line and then passivated with 1-decene via thermal hydrosylation. Once passivated, the SiNC were separated into distinct fractions based on diameter as detailed in our recent publications.^45–47 Size purification was achieved through DGU using a Beckman Coulter centrifuge with a Ti41 rotor. A step gradient of 50 %, 60 %, 70 %, 80 %, and 90 % chloroform in m-xylene was used as the transport and separation medium for the nanocrystals. Nine different spins were conducted with speeds and times ranging from 30k-41k RPM and 10-35 hours, respectively, with the rate of 35k RPM and a time of 24 hours being used for most separations. Solution state PL was determined for the fractions of all nine spins and fractions of comparable peak emission were then combined to form super fractions.

The super fractions were optically characterized in both the solvated and solvent free state, where the QY, PL and lifetime (spectrally resolved and full spectrum) were measured. For all solvent exchange processes carried out herein an inert atmosphere of nitrogen was used to ensure the limited oxidation of the SiNCs. For all QY measurements, an integrating sphere was used with a fiber-coupled 20 mW Omicron PhoxX 375 nm laser for excitation and an Ocean Optics QE65000 spectrometer for detection. The PL measurements from Figure S2 of the supplementary material were also done in this set-up. FS solvent-free lifetime measurements were conducted on a custom inverted microscope with a 60x water immersion objective and an in-house built sample holder using modulated pulsed excitation at 375 nm delivered through a notch filter at a rate of 1-5 kHz (Advanced Laser Diode Systems, PiL037, 30 ps pulse width, 140 mW peak power). A photomultiplier tube was coupled to a digital oscilloscope for detection. FS solution state lifetime measurements were performed using the same excitation and detection sources as the FS solvent-free samples previously mentioned, but on a customized upright microscope with a 4x objective. Delta linear-variable filters (LVFs) with an in-house modified Ocean Optics LVF mount were positioned before the detector for the spectrally resolved measurements. Solution-state SR lifetime measurements were collected using a Horiba Jobin Yvon Fluorolog-3 spectrofluorimeter with a 450-W xenon short-arc excitation source that was mounted vertically in an air-cooled housing. An off-axis mirror was used for light focusing and collection with double-grating excitation and emission Czerny-Turner spectrometers, kinematic classically ruled gratings, and all-reflective optics. The detector was an R928P for high sensitivity in photon-counting mode. For all lifetime measurements, the mean excitation power per unit area was kept well inside the regime of linear PL response. A JEOL JEM-2100 analytical transmission electron microscope operated at 200 kV with a GATAN Orius SC1000 CCD was used for all electron microscopy images.

See supplementary material for details related to size purification, characterization and numerical modeling.

EKH acknowledges the support of the National Science Foundation (NSF) through CBET-1133135 and CBET-1603445, and the US Department of Energy (DOE) through DE-FG36-08GO88160. RK, JS, MPS thank the department of Chemistry and Biochemistry at NDSU for financial support.