On the nature of the stretched exponential photoluminescence decay for silicon nanocrystals
© Zatryb et al; licensee Springer. 2011
Received: 19 September 2010
Accepted: 31 January 2011
Published: 31 January 2011
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© Zatryb et al; licensee Springer. 2011
Received: 19 September 2010
Accepted: 31 January 2011
Published: 31 January 2011
The influence of hydrogen rate on optical properties of silicon nanocrystals deposited by sputtering method was studied by means of time-resolved photoluminescence spectroscopy as well as transmission and reflection measurements. It was found that photoluminescence decay is strongly non-single exponential and can be described by the stretched exponential function. It was also shown that effective decay rate probability density function may be recovered by means of Stehfest algorithm. Moreover, it was proposed that the observed broadening of obtained decay rate distributions reflects the disorder in the samples.
The discovery of visible photoluminescence (PL) from porous silicon and then silicon nanocrystals (Si-NCs) has stimulated a great deal of interest in this material mainly due to a number of promising potential applications, like, for instance, light emitting diodes  or silicon-based lasers . Although the quantum efficiency of Si-NCs emission gives hope for future device applications, it remains low compared to the direct band gap III-V or II-VI materials. It is partially due to technological problems with fabrication of defect-free and structurally uniform Si-NCs samples, where nonradiative recombination sites do not play a key role in the emission process. From this point of view, the improvement of Si-NCs emission quantum efficiency remains an important challenge for further optoelectronic applications. Thus, any experimental tool that leads to information about non-uniformity of Si-NCs structures and its influence on emission properties is valuable.
The optical properties of Si-NCs can be investigated by means of time-resolved spectroscopy. This method often brings new information about disorder in the ensembles of emitters, especially in complex systems where various collective phenomena result in complicated time dependence of the experimental PL decay. Particularly, in the case of Si-NCs different authors have shown [3, 4] that very often PL decay exhibits stretched exponential line shape. However, the physical origin of such behavior remains a matter of discussion. For this moment, a few explanations have been given by different authors, such as exciton migration between interconnected nanocrystals , variation of the atomic structure of Si-NCs of different sizes , carriers out tunneling from Si-NCs to distribution of nonradiative recombination traps  and many other [8, 9]. Therefore, it is still important to gather some new experimental evidence in this field.
It should be also emphasized that in the case of stretched exponential relaxation function, the PL decay may be analyzed more thoroughly by recovering the distribution of recombination rates . Surprisingly, this kind of approach is very rare with Si-NCs, especially for structures deposited by the sputtering method. Very few papers report on recombination rate distributions calculated by means of inverse Laplace transform  for porous silicon or by means of the maximum entropy method for Si-NCs produced by laser pyrolysis of silane . Therefore, it is worth investigating the evolution of such distributions also in the case of other deposition methods.
In this work, we study the absorption properties as well as PL decays measured for Si-NCs thin films deposited by the magnetron sputtering method. It is shown that time dependence of PL may be described by the stretched exponential function. The distributions of recombination rates are calculated numerically by means of inverse Laplace transform. The influence of structural disorder on carrier relaxation kinetics is discussed.
The silicon-rich-silicon oxide (SRSO) films with a nominal thickness of 500 nm used for this study were deposited onto quartz substrates by radio-frequency reactive magnetron sputtering. The incorporation of Si excess was monitored through the variation of the hydrogen rate r H = P H2/(P Ar + P H2) from 10% to 50%. The films were deposited without any intentional heating of the substrates and with a power density of 0.75 W/cm2. All samples were subsequently annealed at 1,100°C for 1 h under N2 flux in order to favor the precipitation of Si excess and to induce Si-NCs formation.
The absorption properties were investigated by means of transmission and absorption measurements (with mixed xenon and halogen light sources). Time-resolved photoluminescence spectra were investigated by means of strobe technique with pulsed xenon lamp used as an excitation source and photomultiplier tube (PMT) used for detection. For the excitation wavelength used in our experiment (350 nm) the pulse width at half maximum was about 2 μs. To calculate the inverse Laplace transform for decay rates recovery, numerical calculations were performed using Stehfest algorithm  (for N = 14).
In our previous papers [13, 14] detailed structural investigations (including atomic force microscopy, X-ray diffraction, high-resolution electron microscopy or Rutherford backscattering) of SRSO films fabricated with the same technological conditions have been reported. The main conclusion of these investigations has shown that the increase of r H used during deposition leads to increased disorder in the sample. Namely, deposition with r H = 10% favors formation of well-crystallized Si-NCs with average size of about 3 nm, whereas deposition with r H = 50% favors formation of mostly amorphous Si nanograins with size less than 2 nm.
What is more, below the optical band gap, the spectra shown in Figure 2 reveal long, exponentially decreasing absorption edges. Assuming that these absorption tails have amorphous nature  related to the structural non-uniformity of samples, they can be approximately described by Urbach equation: α = C exp(E/E U), where E U is the characteristic Urbach energy determining the exponential slope. Figure 2 clearly shows that E U increases with increasing r H, being of the order of hundreds of millielectron volt. Longer tails in the absorption edge for higher r H constitute another experimental evidence for stronger disorder in samples with higher r H.
To analyze the influence of structural disorder on emitters relaxation kinetic, we introduce a time-domain relaxation function that describes how system returns to equilibrium after a perturbation. Namely, after light illumination we obtain some number of emitters n(0) in the excited state. When we turn off the illumination the system returns to equilibrium after some time. Thus, our relaxation function may be defined as time-dependent excited emitters fraction n(t)/n(0).
where τ 0 is an effective time constant and β is a constant between 0 and 1.
where C is a constant.
We would like to emphasize that Eq. 6 may be used to model many kinds of non-single exponential PL decays. This is an important issue, since in many papers the photoluminescence decay curve, measured in a similar way as described in the text, is modeled with the time dependence of Eq. 1. While this is a very good method to quantitatively describe the extent to which the decay is non-single exponential, it does not provide direct information about the Φ(k) probability density function.
where Γ is the Gamma function.
In the investigated case, it was found that the β constants were equal to 0.68, 0.57, 0.55 for r H = 10%, 30%, 50%, respectively. The average decay times < τ> (and τ 0) were equal to 70 μs (τ 0 ≈ 54 μs), 48 μs (τ0 ≈ 30 μs) and 36 μs (τ0 ≈ 21 μs) for r H = 10%, 30%, 50%, respectively. Both parameters decrease with increasing r H. Moreover, the values of β and τ remain close to already reported results .
where a = β(1 - β)- 1 and τ = τ 0[β(1-β)1/ a ]- 1.
Alternatively, the inversion of Laplace transform (Eq. 8) may be computed numerically using Stehfest algorithm . By comparing the numerical inversion of Eq. 8 with Eq. 9, we obtained the same results for Φ(k) distribution. In this way, we have found very good Stehfest algorithm accuracy for this class of functions. This last result may be important in a case when n(t)/n(0) function is a bit different than stretched-exponential, calculations with Stehfest algorithm should bring good results.
To explain the observed features, it should first be mentioned that the obtained Φ(k) function provide information about both the radiative (k R) and nonradiative (k NR) relaxation rates . However, a very low quantum efficiency of Si-NCs emission suggests that nonradiative processes should be predominant (k NR >> k R). This allows us to relate the changes observed in the decay rate distribution Φ(k) to the introduction of more defect states to the matrix containing Si-NCs after increasing the r H factor. These new states may act as nonradiative recombination paths for the excited carriers , leading to broadening of Φ(k) function and shortening of the average decay time (Figure 4b).
It is noteworthy that the above interpretation is also consistent with the rest of experimental results. As it was mentioned at the beginning, increasing r H results in higher structural disorder, which, in turn, may be the reason behind the appearance of new nonradiative states and the simultaneous increase of the Urbach energy E U (see the inset to Figure 4b). What is more, this interpretation may be also supported by our recent results  obtained for multilayered SRSO films with Si-NCs. In this work, we investigated samples with constant Si-NCs size co-doped with different amounts of boron. We have found that introduction of these impurities to Si-NCs environment leads to stronger deviation from single-exponential PL decays, which was interpreted as a result of appearance of new nonradiative sites. This result also correlates to the model proposed by Suemoto et al. , where broad distributions of decay rates were interpreted as a result of different potential barriers for carriers out-tunneling from Si-NCs to nonradiative sites.
On the other hand, it has been shown  that excitation may migrate from nanocrystals to light-emission centers, such as S = O bonds. Thus, if probability of such migration depends on nanocrystal structure (size or crystallinity), it is also possible that radiative recombination centers responsible for emission at 1.5 eV have a broad k R distribution (because, as structural results have shown, size and crystallinity changes with r H). Therefore, if k R was comparable with k NR then the shape of Φ(k) function could be influenced somewhat by the k R distribution. This should be especially important for samples with high quantum efficiency of Si-NCs emission (where k NR << k R or both rates are comparable), which is not the case. Nevertheless, it is worth noting that in such case, it is also possible to obtain a broad Φ(k) probability density function and stretched-exponential PL decay.
To sum up, it has been shown that PL decay of Si-NCs is strongly non-single exponential and may be described by stretched exponential function. It has been demonstrated that effective decay rate probability density function may be recovered with very good accuracy by means of numerical inversion of the Laplace transform (using Stehfest algorithm) as well as using asymptotic function. In this way, broad decay rate distributions were obtained. It has been proposed that the observed broadening of the distributions and the decrease of average decay time constant for higher r H factors is related to the appearance of more nonradiative states in the Si-NCs environment.
The authors would like to offer their sincere gratitude to Prof. K. Weron for her valuable insight into the theoretical explanation of the relaxation processes. What is more, A. P. would like to acknowledge for financial support to Iuventus Plus program (no. IP2010032570).
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