On the nature of the stretched exponential

photoluminescence decay for silicon nanocrystals

G Zatryb, A Podhorodecki, J Misiewicz, J. Cardin, F Gourbilleau

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G Zatryb, A Podhorodecki, J Misiewicz, J. Cardin, F Gourbilleau. On the nature of thestretched exponential photoluminescence decay for silicon nanocrystals. Nanoscale ResearchLetters, SpringerOpen, 2011, 6 (106), pp.1-8. <10.1186/1556-276X-6-106>. <hal-01139278>

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NANO EXPRESS Open Access

On the nature of the stretched exponentialphotoluminescence decay for silicon nanocrystalsG Zatryb1, A Podhorodecki1*, J Misiewicz1, J Cardin2, F Gourbilleau2

Abstract

The influence of hydrogen rate on optical properties of silicon nanocrystals deposited by sputtering method wasstudied by means of time-resolved photoluminescence spectroscopy as well as transmission and reflectionmeasurements. It was found that photoluminescence decay is strongly non-single exponential and can bedescribed by the stretched exponential function. It was also shown that effective decay rate probability densityfunction may be recovered by means of Stehfest algorithm. Moreover, it was proposed that the observedbroadening of obtained decay rate distributions reflects the disorder in the samples.

IntroductionThe discovery of visible photoluminescence (PL) fromporous silicon and then silicon nanocrystals (Si-NCs)has stimulated a great deal of interest in this materialmainly due to a number of promising potential applica-tions, like, for instance, light emitting diodes [1] or sili-con-based lasers [2]. Although the quantum efficiency ofSi-NCs emission gives hope for future device applica-tions, it remains low compared to the direct band gapIII-V or II-VI materials. It is partially due to technologi-cal problems with fabrication of defect-free and structu-rally uniform Si-NCs samples, where nonradiativerecombination sites do not play a key role in the emis-sion process. From this point of view, the improvementof Si-NCs emission quantum efficiency remains animportant challenge for further optoelectronic applica-tions. Thus, any experimental tool that leads to informa-tion about non-uniformity of Si-NCs structures and itsinfluence on emission properties is valuable.The optical properties of Si-NCs can be investigated

by means of time-resolved spectroscopy. This methodoften brings new information about disorder in theensembles of emitters, especially in complex systemswhere various collective phenomena result in compli-cated time dependence of the experimental PL decay.Particularly, in the case of Si-NCs different authors haveshown [3,4] that very often PL decay exhibits stretched

exponential line shape. However, the physical origin ofsuch behavior remains a matter of discussion. For thismoment, a few explanations have been given by differ-ent authors, such as exciton migration between inter-connected nanocrystals [5], variation of the atomicstructure of Si-NCs of different sizes [6], carriers outtunneling from Si-NCs to distribution of nonradiativerecombination traps [7] and many other [8,9]. There-fore, it is still important to gather some new experimen-tal evidence in this field.It should be also emphasized that in the case of

stretched exponential relaxation function, the PL decaymay be analyzed more thoroughly by recovering the distri-bution of recombination rates [10]. Surprisingly, this kindof approach is very rare with Si-NCs, especially for struc-tures deposited by the sputtering method. Very few papersreport on recombination rate distributions calculated bymeans of inverse Laplace transform [7] for porous siliconor by means of the maximum entropy method for Si-NCsproduced by laser pyrolysis of silane [11]. Therefore, it isworth investigating the evolution of such distributions alsoin 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 depos-ited by the magnetron sputtering method. It is shownthat time dependence of PL may be described by thestretched exponential function. The distributions ofrecombination rates are calculated numerically bymeans of inverse Laplace transform. The influence ofstructural disorder on carrier relaxation kinetics isdiscussed.

* Correspondence: artur.p.podhorodecki@pwr.wroc.pl1Institute of Physics, Wroclaw University of Technology, WybrzezeWyspianskiego 27, 50-370 Wroclaw, PolandFull list of author information is available at the end of the article

Zatryb et al. Nanoscale Research Letters 2011, 6:106http://www.nanoscalereslett.com/content/6/1/106

© 2011 Zatryb et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons AttributionLicense (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium,provided the original work is properly cited.

Experimental detailsThe silicon-rich-silicon oxide (SRSO) films with a nom-inal thickness of 500 nm used for this study were depos-ited onto quartz substrates by radio-frequency reactivemagnetron sputtering. The incorporation of Si excesswas monitored through the variation of the hydrogenrate rH = PH2/(PAr + PH2) from 10% to 50%. The filmswere deposited without any intentional heating of thesubstrates and with a power density of 0.75 W/cm2. Allsamples were subsequently annealed at 1,100°C for 1 hunder N2 flux in order to favor the precipitation of Siexcess and to induce Si-NCs formation.The absorption properties were investigated by means of

transmission and absorption measurements (with mixedxenon and halogen light sources). Time-resolved photolu-minescence spectra were investigated by means of strobetechnique with pulsed xenon lamp used as an excitationsource and photomultiplier tube (PMT) used for detec-tion. For the excitation wavelength used in our experiment(350 nm) the pulse width at half maximum was about 2μs. To calculate the inverse Laplace transform for decayrates recovery, numerical calculations were performedusing Stehfest algorithm [12] (for N = 14).

Results and discussionIn our previous papers [13,14] detailed structural inves-tigations (including atomic force microscopy, X-ray dif-fraction, high-resolution electron microscopy orRutherford backscattering) of SRSO films fabricatedwith the same technological conditions have beenreported. The main conclusion of these investigationshas shown that the increase of rH used during deposi-tion leads to increased disorder in the sample. Namely,deposition with rH = 10% favors formation of well-crys-tallized Si-NCs with average size of about 3 nm, whereasdeposition with rH = 50% favors formation of mostlyamorphous Si nanograins with size less than 2 nm.Figure 1 shows time-resolved PL spectra measured for

samples with rH = 10%, 30% and 50%. In each case, thebroad emission band centered at around 1.5 eV may beobserved (the nonsymmetrical emission band shape isdue to the cut-off of PMT detector). What is more, thePL intensity significantly drops after increasing rH from10% to 30% and then decreases only slightly. The lackof PL shift with rH variation indicates that the observedemission rather cannot be attributed to the quantumconfinement effect. It may, however, be attributed tosome emission centers at the interface between Si-NCsand SiO2 matrix (surface states) because according towhat has been shown [15], defect states localized on thenanocrystal surface may suppress the quantum confine-ment effect on the emission spectra. Similar effect has

also been observed in one of our previous papers [16]for Si-NCs.Figure 2 shows the absorption (a) spectra calculated

from reflectance (R) and transmittance (T) measure-ments according to the equation a ∝ -ln(T/(1 - R)2)which allows us to deal with disturbing interference pat-terns [17]. For the higher energy part of the spectra, theTauc formula (aE) = A(E - Eg)

m was used to estimatethe optical band gap (Eg). The best fit to the experimen-tal data was obtained for m = 1/2, which corresponds todirect allowed transition. It may also be noted that theabsorption edge is significantly blue-shifted from 3.76eV for rH = 10% to 4.21 eV for rH = 50%. The observedblue-shift of absorption edge may be related to quantumconfinement effect which was discussed by us in moredetails elsewhere [16].What is more, below the optical band gap, the spec-

tra shown in Figure 2 reveal long, exponentiallydecreasing absorption edges. Assuming that theseabsorption tails have amorphous nature [18] related tothe structural non-uniformity of samples, they can beapproximately described by Urbach equation: a = Cexp(E/EU), where EU is the characteristic Urbachenergy determining the exponential slope. Figure 2clearly shows that EU increases with increasing rH,being of the order of hundreds of millielectron volt.Longer tails in the absorption edge for higher rH con-stitute another experimental evidence for stronger dis-order in samples with higher rH.To analyze the influence of structural disorder on

emitters relaxation kinetic, we introduce a time-domainrelaxation function that describes how system returns toequilibrium after a perturbation. Namely, after light illu-mination we obtain some number of emitters n(0) inthe excited state. When we turn off the illumination thesystem returns to equilibrium after some time. Thus,our relaxation function may be defined as time-depen-dent excited emitters fraction n(t)/n(0).It is well known [19] that for an ensemble of emitters,

the relaxation function may be described by Laplacetransform of some non-negative function F(k):

n t

nkt k dk

00

exp (1)

In Equation 1, the function F(k) may be interpreted asan effective decay rate probability density function.Therefore, the decay rate k is in fact a positive randomvariable. It can be shown [20] that if relaxation of theexcited emitters to the ground state may occur throughmany competing channels (for example the excited

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carriers may escape from nanocrystal to many differentradiative or nonradiative centers), the relaxation func-tion is given by stretched-exponential (Kohlrausch)function:

n t

n

t

0 0

exp

(2)

where τ0 is an effective time constant and b is a con-stant between 0 and 1.However, in our experiment we do not measure the

relaxation function directly. Instead, we measure thenumber of photons NPh emitted in a very short timeperiod δt after the excitation pulse. Using a delay gategenerator, we sweep the delay between moment of mea-surement and the excitation pulse, creating the PLdecay curve. Because NPh is directly proportional to the

change of excited emitters number Δn = n(t’+δt) - n(t’),we may define the decay of PL intensity as a negativetime derivative of the relaxation function:

I tn

dn t

dtPL 1

0(3)

Equation 3 is in fact a definition of the so calledresponse function which determines the rate at whichthe relaxation function changes. Thus, in our case, theadequate form of the stretched-exponential function forPL decay is:

I t C tt

PL

1

0

exp (4)

where C is a constant.

Figure 1 The time-resolved PL measured for samples with various rH (10%, 30% and 50%). The nonsymmetrical emission band shape isdue to the cutoff of PMT detector.

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It is also worth noting that Eq. 3 and Eq. 1 imply amore general relation between F(k) function and PLdecay, namely:

I td

dtkt k dkPL

exp 0

(5)

which in turn leads to the following expression:

I t k kt k dkPL

exp 0

(6)

We would like to emphasize that Eq. 6 may be usedto model many kinds of non-single exponential PLdecays. This is an important issue, since in manypapers the photoluminescence decay curve, measuredin a similar way as described in the text, is modeledwith the time dependence of Eq. 1. While this is a verygood method to quantitatively describe the extent to

which the decay is non-single exponential, it does notprovide direct information about the F(k) probabilitydensity function.Figure 3 shows experimental PL decays measured at

around 820 nm (PL peak, 1.5 eV). Hundreds-microse-conds long, strongly non-single exponential decay pro-files were obtained that can be well described by Eq. 4.The least-squares fit of the Eq. 4 to experimental databrings values of τ0 and b. Having both constants, it ispossible to define average decay time constant <τ> inthe following form:

0 1 (7)

where Γ is the Gamma function.In the investigated case, it was found that the b constants

were equal to 0.68, 0.57, 0.55 for rH = 10%, 30%, 50%,respectively. The average decay times <τ> (and τ0) wereequal to 70 μs (τ0 ≈ 54 μs), 48 μs (τ0 ≈ 30 μs) and 36 μs(τ0 ≈ 21 μs) for rH = 10%, 30%, 50%, respectively. Both

Figure 2 Absorption spectra calculated on the basis of the reflectance and transmittance measurements. The left axis shows longexponential tails in the absorption edge. The right axis shows curves used for estimating the band gap according to the Tauc formula. The blue-shift of the absorption edge may be observed with increasing rH.

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parameters decrease with increasing rH. Moreover, thevalues of b and τ remain close to already reportedresults [21].To analyze stretched exponential behavior in more

details, we may recover the decay rates probability den-sity function F(k). Using an analytical expression, suchas Eq. 2 with b and τ0 taken from the experimental datafit to Eq. 4, it is possible to recover F(k) by means ofinverse Laplace transform, solving the following equation:

Ln t

nk

1

0 (8)

Obviously, the Eq. 7 solution depends on n(t)/n(0)function. In particular, for n(t)/n(0) given by Eq. 2, thereis no general analytical solution and only asymptoticform of F(k) distribution may be obtained by the sad-dle-point method [22]:

ka

k ka a

2

1 2/exp (9)

where a = b(1 - b)-1 and τ = τ0[b(1-b)1/a]-1.Alternatively, the inversion of Laplace transform (Eq.

8) may be computed numerically using Stehfest algo-rithm [12]. By comparing the numerical inversion of Eq.8 with Eq. 9, we obtained the same results for F(k) dis-tribution. In this way, we have found very good Stehfestalgorithm accuracy for this class of functions. This lastresult may be important in a case when n(t)/n(0) func-tion is a bit different than stretched-exponential, calcu-lations with Stehfest algorithm should bring goodresults.Figures 4a,b show decay rate distribution F(k) calcu-

lated from Eq. 8. As expected, a power-like dependencemay be observed (Figure 4a) for high values of theabscissa variable. The obtained distributions are very

Figure 3 The non-single exponential PL decays measured for samples with different rH. The solid line stands for stretched exponentialfunction fit (Eq. 1). b parameters and the average time constant <τ> are shown.

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Figure 4 Effective decay rate probability density functions. Power-like dependence may be observed for higher decay rates (a) in log-logscale. The distribution broaden significantly with the increase of rH. Shift of the distribution is visible (b) together with average decay time dropand EU rise for higher rH (inset). The normalization in (b) was carried out in a manner exposing distributions broadening while in (a) the functionF(k) is properly normalized probability density function.

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broad with long tails directed towards shorter lifetimes,which demonstrates the strongly non-single exponentialcharacter of decay curves. While increasing rH, thedecay rate distribution F(k) shifts towards higher decayrates (Figure 4b). What is more, F(k) broaden signifi-cantly with increasing rH parameter.To explain the observed features, it should first be

mentioned that the obtained F(k) function provideinformation about both the radiative (kR) and nonradia-tive (kNR) relaxation rates [10]. However, a very lowquantum efficiency of Si-NCs emission suggests thatnonradiative processes should be predominant (kNR

>>kR). This allows us to relate the changes observed inthe decay rate distribution F(k) to the introduction ofmore defect states to the matrix containing Si-NCs afterincreasing the rH factor. These new states may act asnonradiative recombination paths for the excited car-riers [7], leading to broadening of F(k) function andshortening of the average decay time (Figure 4b).It is noteworthy that the above interpretation is also

consistent with the rest of experimental results. As itwas mentioned at the beginning, increasing rH results inhigher structural disorder, which, in turn, may be thereason behind the appearance of new nonradiative statesand the simultaneous increase of the Urbach energy EU(see the inset to Figure 4b). What is more, this interpre-tation may be also supported by our recent results [23]obtained for multilayered SRSO films with Si-NCs. Inthis work, we investigated samples with constant Si-NCssize co-doped with different amounts of boron. We havefound that introduction of these impurities to Si-NCsenvironment leads to stronger deviation from single-exponential PL decays, which was interpreted as a resultof appearance of new nonradiative sites. This result alsocorrelates to the model proposed by Suemoto et al. [24],where broad distributions of decay rates were inter-preted as a result of different potential barriers for car-riers out-tunneling from Si-NCs to nonradiative sites.On the other hand, it has been shown [25] that excita-

tion may migrate from nanocrystals to light-emissioncenters, such as S = O bonds. Thus, if probability of suchmigration depends on nanocrystal structure (size or crys-tallinity), it is also possible that radiative recombinationcenters responsible for emission at 1.5 eV have a broadkR distribution (because, as structural results have shown,size and crystallinity changes with rH). Therefore, if kRwas comparable with kNR then the shape of F(k) functioncould be influenced somewhat by the kR distribution.This should be especially important for samples withhigh quantum efficiency of Si-NCs emission (where kNR

<<kR or both rates are comparable), which is not thecase. Nevertheless, it is worth noting that in such case, itis also possible to obtain a broad F(k) probability densityfunction and stretched-exponential PL decay.

ConclusionsTo sum up, it has been shown that PL decay of Si-NCsis strongly non-single exponential and may be describedby stretched exponential function. It has been demon-strated that effective decay rate probability density func-tion may be recovered with very good accuracy bymeans of numerical inversion of the Laplace transform(using Stehfest algorithm) as well as using asymptoticfunction. In this way, broad decay rate distributionswere obtained. It has been proposed that the observedbroadening of the distributions and the decrease of aver-age decay time constant for higher rH factors is relatedto the appearance of more nonradiative states in the Si-NCs environment.

AcknowledgementsThe authors would like to offer their sincere gratitude to Prof. K. Weron forher valuable insight into the theoretical explanation of the relaxationprocesses. What is more, A. P. would like to acknowledge for financialsupport to Iuventus Plus program (no. IP2010032570).

Author details1Institute of Physics, Wroclaw University of Technology, WybrzezeWyspianskiego 27, 50-370 Wroclaw, Poland 2CIMAP, UMR CNRS/CEA/ENSICAEN/UCBN, Ensicaen 6 Bd Maréchal Juin, 14050 Caen Cedex 4, France

Authors’ contributionsGZ, AP and JM carried out the spectroscopic measurements as well ascalculations. JC and FG designed and deposited the investigated samples.All authors read and approved the final manuscript.

Competing interestsThe authors declare that they have no competing interests.

Received: 19 September 2010 Accepted: 31 January 2011Published: 31 January 2011

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doi:10.1186/1556-276X-6-106Cite this article as: Zatryb et al.: On the nature of the stretchedexponential photoluminescence decay for silicon nanocrystals.Nanoscale Research Letters 2011 6:106.

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