We investigate the spontaneous emission decay rate of Si nanocrystals modified by thin semicontinuous gold films. It has been shown that the mean and standard deviation values of decay rate distribution obtained from the photo-emission decay curve analysis increase due to the deposition of semicontinuous gold films. These values are dependent on gold film thickness and emission wavelength. The observed results are well explained using a point-dipole decay rate model considering the effective dielectric functions of the gold films that exhibit peculiar structures in the localized surface plasmon resonance region.
© 2012 OSA
The spontaneous emission decay rate of a luminescent material can be changed by depositing metal nanostructures on its surface. This change is known to be induced by the electromagnetic interactions resulting from the excitation of surface plasmons in the nanostructured metal by the luminescent material. Metal nanostructures can be classified into two types based on their geometry: planar metal films and non-planar metal nanostructures such as nanoparticle arrays and thin semicontinuous metal films consisting of disorder-shaped nanostructures. When a planar metal film is placed above a luminescent material at a small distance, the emission decay rate of the luminescent material increases due to the excitation of the propagating-mode surface plasmons [1, 2]. This phenomenon can be successfully interpreted using a theoretical model based on point dipole approximation [3, 1]. In non-planar metal nanostructures, localized modes of the surface plasmon play an important role in changing the decay rate of a luminescent material. Therefore, the decay rate characteristics for non-planar metal nanostructures are quite different from those for planar metal films, e.g., strong dependence of the decay rate on wavelength  and polarization  and fluctuation of decay rate distribution . Note that the fluctuation of decay rate has been observed for an emitter near dielectric scatterers.  However, there are few theoretical models to describe the change in the decay rates due to non-planar metal nanostructures.
Changes in the photoluminescence (PL) intensity and the spontaneous emission decay rate due to the deposition of metal nanostructures have been observed in a wide variety of luminescent materials, e.g., semiconductor nanocrystals and organic molecules [8, 9, 10]. It has been demonstrated that the PL intensity of Si nanocrystals can be considerably enhanced using non-planar metal nanostructures [11, 12, 13]. The photo-emission decay curves of Si nanocrystals on a planar metal film were studied [14, 15, 16] and the decay rate increased due to the excitation of surface plasmons in the planar metal film was observed . However, there is no detailed experimental study on the effect of deposited non-planar metal nanostructures on the emission decay rate of Si nanocrystals. This may in part be due to the exceptionally difficult process of analyzing the emission decay curves of Si nanocrystals .
In this work, we investigate the emission decay rate of Si nanocrystals near thin semicontinuous gold film with different mass thicknesses. We obtain the decay rate distribution by analyzing emission decay curves over a wide wavelength range. The thickness and wavelength dependence of the mean and standard deviation values of the decay rate distribution are well explained by a theoretical calculation based on the macroscopic effective dielectric functions of gold films that reveal the localized surface plasmon resonance peak.
We used porous silicon (PSi), which can be regarded as an assembly of Si nanocrystals. A 75-nm-thick PSi layer was prepared by conventional anodic etching. Thin Au layers were then deposited on the PSi layer by vacuum evaporation. The mass thicknesses of the deposited Au layers were d = 1.0, 2.5, and 5.0 nm. Figure 1(a) shows a scanning electron microscope (SEM) image of a Au layer (d = 5 nm) deposited by vacuum evaporation. As understood from this image, the deposited Au layer is semicontinuous and consists of nano-islands. A nanostructured Au layer, such as the one described above, has a broad absorption band in the red region due to the resonance excitation of localized surface plasmons in Au [18, 19]. Figure 1(b) shows the absorbance spectra of Au films. Clear absorbance peaks are observed in the 600–750 nm wavelength range with dependence on film thickness. As can be seen from Fig. 1(c), the PL intensity of the PSi layer decreases with increasing Au layer thickness. We measured wavelength-dependent emission decay curves of these PSi/Au samples in the 580–740 nm wavelength range. Decay curve measurements were performed using a frequency-tripled 355-nm light pulse from a Nd:YAG laser (pulse duration: 5 ns). The excitation power density is ∼25 mW/cm2. Photo-emission signals were recorded using a grating spectrometer (JASCO CT-25C), a Peltier-device-cooled photomultiplier tube (Hamamatsu R375), and a multichannel scaler (Stanford Research SR430). The overall time resolution of the system was better than 100 ns. All the measurements were performed at room temperature. Spectroscopic ellipsometry (SE) measurements were also performed using a spectroscopic phase modulated ellipsometer (HORIBA Jobin Yvon, UVISEL NIR) to determined the optical response of the PSi/Au samples.
3. Results and discussion
Figure 2(a) shows the emission decay curves of PSi (d = 0 nm) and the PSi/Au samples. The decay curves are found to be strongly modified by the deposition of Au films, i.e., the decay rate increases with the thickness of the Au layer. Given that our experimental decay curves exhibit a non-single-exponential feature, we fitted these decay data using two different models. One is the stretched-exponential decay model, I(t) = I0(β/t)(wstt)β exp[−(wstt)β] [20, 21], where I0, wst, and β are the pre-exponential factor, the decay rate, and the stretched factor, respectively. This model is widely used for the decay curve analysis of Si nanocrystals [22, 23, 14]. The other model used was the one proposed by van Driel et al. , which takes into consideration the lognormal distribution of decay rates:Eq. (2), A is the normalization constant and wmf is the most frequent rate constant. Both these models required the use of three fitting parameters: I0, wst, and β (stretched-exponential decay model) and I0, wmf, and wd (lognormal-distribution decay model).
The fitted decay curves are shown in Fig. 2(a) (dashed curves: stretched exponential model; solid curves: lognormal-distribution decay model). For the d = 0 nm sample (PSi), the goodness-of-fit achieved with the lognormal decay model is better than that with the stretched-exponential decay model (the reduced χ square values  for the lognormal and the stretched-exponential decay models are 0.73 and 3.4, respectively). For the PSi/Au samples, the lognormal decay model gives a better fit with the experimental data ( and 0.74 for d = Au 1.0 and 2.5 nm, respectively) than does the stretched-exponential decay model ( and 26 for d = 1.0 and 2.5 nm, respectively). Note that the better fit with the lognormal decay model is achieved over the entire wavelength range investigated. This indicates that the log-normal decay model is more reliable in the analysis of decay curves in metal/Si-nanocrystal systems. In the following, we present the results obtained using the lognormal-distribution decay model.
Figure 2(b) shows the decay rate distributions, ρ(w), of PSi and Au/PSi samples (d = 1.0 and 2.5 nm) calculated using the following fit-determined parameters: wmf = 0.042 (PSi), 0.12 (1.0 nm), and 0.19 μs−1 (2.5 nm); wd = 0.094 (PSi), 0.40 (1.0 nm), and 0.71 μs−1 (2.5 nm), where wmf and wd correspond to the peak value and width of ρ(w), respectively. These results show that ρ(w) can be strongly modified upon the deposition of a Au layer, i.e., the peak in ρ(w) shifts toward the larger decay-rate side and has larger width in distribution.
The peak shift in ρ(w) can be qualitatively explained as follows: Because Si nanocrystals are located close to the Au layer, the surface plasmons in this layer can be excited. This results in the creation of additional radiative and/or nonradiative decay processes. Owing to surface plasmon excitation, therefore, the total decay rate wt increases according to25]. This increase in wt results in the peak shift of ρ(w) toward the larger decay-rate side.
The peak shift of ρ(w) becomes more pronounced for the sample with thicker Au layer. This is to be due to the difference in the surface plasmon resonance wavelength. As shown in Fig. 1(b), the surface plasmon resonance occurs at ∼600 and ∼670 nm with the Au layer thicknesses of 1.0 and 2.5 nm, respectively. Thus, because the resonant surface plasmon excitation occurs near 670 nm, the peak shift in ρ(w) measured at 660 nm becomes large for the sample with Au layer thickness of 2.5 nm.
The mean value wmean of decay rate distribution can be obtained from the fit-determined parameters wmf and wd :Fig. 2(b)]. Figure 3(a) shows wmean as a function of emission wavelength λ for various Au thicknesses. The wmean of PSi/Au was normalized to that of PSi. This parameter increases with increasing λ, and shows a higher value for samples with thicker Au layers. The maximum wmean value is ∼30 at λ = 740 nm for the sample with d = 5.0 nm.
To quantitatively explain the increase in the spontaneous emission decay rate due to the Au deposition and its wavelength dependence, we used a theory developed by Chance, Procke, and Silbey (CPS model) . The decay rate of a classical oscillating dipole in a layered system consisting of metal and dielectric media can be calculated under the following assumptions [1, 3]: (i) The size of the dipole is small compared to the emission wavelength (i.e., point dipole approximation), (ii) interface between each layer is flat, (iii) metal and dielectric media are continuous, and (iv) optical response of the media can be completely described by the macroscopic dielectric function. According to the CPS model, spontaneous emission decay rate γ̂ of an isotropic dipole emitter in an arbitrary layered system normalized by the decay rate in vacuum is given by the equation ,3. As evidenced from Eqs. (5) – (7), the decay rate γ̂ is calculated from and .
In the above calculation, we used a five-layer model consisting of a substrate (Si), an emission layer (PSi), a native oxide layer (SiO2), metal layer (Au), and ambient (air), as schematically shown in Fig. 3(b). The dielectric functions of Si and SiO2 were taken from the literature , whereas the dielectric functions of PSi and Au layers were directly determined from SE measurements. In the analysis of SE data, PSi layer is treated as a homogeneous composite layer consisting of air and silicon. Such an effective medium approximation is commonly used for various PSi samples to estimate the optical constants of porous layers.  In our study, the Au layer is a semicontinuous film rather than a continuous film [Fig. 1(a)]. Note that the surface of PSi layer has a complicated structure due to its porous nature. Therefore, to simplify the SE analysis and to satisfy assumptions (ii), (iii), and (iv) in the CPS model, we treated the semicontinuous Au layer as a homogeneous mixture of Au metal and other components (SiO2, air, etc.) with an effective modeled thickness. By using such a simplified analysis of ellipsometry data, anomalous absorption spectra due to the excitation of localized surface plasmons in semicontinuous films were successfully explained [29, 30]. In the present analysis, we used the effective dielectric functions having an effective modeled thickness of ∼9 nm, which we determined by analyzing the measured SE data.
The real (εRe) and imaginary (εIm) parts of the effective dielectric functions of Au layers are shown in Fig. 3(c). The strong dependences of the dielectric functions both on the thickness (d) of the Au layer and wavelength are obvious. For Au layers of d = 1.0 and 2.5 nm, εRe values are positive over the whole wavelength region, while for d = 5.0 nm it is negative in the wavelength range from 500 to 750 nm. This different sing of εRe indicates that the macroscopic optical properties of the Au layers change from insulator-like to metallic with increasing d. Such a change in the optical properties has also been observed in thin Au films of d = 3 – 10 nm .
For supporting surface plasmon modes, metal and its surrounding medium principally have an opposite sign in εRe. We note, however, that the positive sign of εRe for the samples with d = 1.0 and 2.5 nm does not mean no excitation of the surface plasmon modes in such very thin Au-deposited samples. Our obtained εRe reflects only a macroscopic optical response of the effective metal/dielectric layers and therefore results in positive εRe values. When the size of the metal particle becomes smaller than the mean free path of the free electrons, the electron collides with the boundary of the particle. If we consider such size effects, the real dielectric functions εRe of Au particles with diameters still smaller than 5 nm show negative sign in the wavelength region of ≥500 nm with values almost the same as those of bulk Au . Thus, it is not surprising to observe surface plasmon polaritons in a thin semicontinuous Au film. Scholl et al. have also observed plasmon resonances of individual Ag nanoparticles . As the Ag nanoparticle diameter decreased from 20 nm to less than 2 nm, the plasmon resonance shifted to higher energy by 0.5 eV. These authors also presented an analytical quantum mechanical model that describes this shift due to a change in particles dielectric permittivity.
In Fig. 3(c), peaks in εIm are observed in the range from 550 to 800 nm. These peaks correspond to the resonance of the ensemble mode of localized surface plasmons excited in each Au nanoparticle, which is so-called optical conduction resonance [33, 34]. The optical conduction resonance was popularly observed in metal/dielectric composites  and metal island films , and its energy position and strength were well described by the Maxwell-Garnett effective medium theory . Such ensemble mode of localized surface plasmons is accompanied with enhanced local electromagnetic fields near the metal layer and thus results in the enhancement of Raman scattering  and luminescence intensities . As can be seen in Fig. 3(c), the resonance peak wavelength shifts toward the longer wavelength side with increasing d in a manner essentially the same as the absorbance peaks observed in Fig. 1(b). The increased εIm peak value with increasing d also corresponds to the increased localized surface plasmon density due to larger density of the metal nanostructures. The red shift in resonance energy may also be caused by the increased size of the deposited Au nanostructures and enhanced electromagnetic coupling of the neighboring plasmonic polaritons.Fig. 3(a)]. The thickness of native oxide layer, t, was determined to be t = 1.0 – 1.2 nm from analyzing the experimental decay rate data.
In Fig. 3(a), solid curves denote the calculated decay rates. We can see good agreement between the calculated and experimental values. This agreement indicates the validity of our calculation model and supports the conjecture that the increase in decay rate with increasing deposited Au thickness is mainly a result of the excitation of localized surface plasmons in the Au nanostructures. The larger increase in the decay rate of the sample with the thicker Au film is also understood to be due to the larger peak value of εIm (i.e., larger optical absorption by surface plasmon excitation), as shown in Fig. 3(c).
We also determined the quantum efficiency, q, in Eq. (5) as a function of emission wavelength. In the inset of Fig. 3(a), q is plotted as a function of λ. As shown, q decreases with increasing λ. This tendency is very similar to that obtained from Si nanocrystals embedded in a SiO2 matrix . However, our q values are considerably smaller than those in the literature (e.g., q = 11% at λ = 740 nm for PSi [the inset of Fig. 3(a)], whereas q = 44% at λ = 760 nm for Si nanocrystals in SiO2 ). This may be due to the non-radiative recombination centers formed on the PSi surface.
The radiative (wr) and nonradiative decay rates (wnr) of PSi can be obtained using γ and q values. The experimentally obtained total decay rate wt is represented in Eq. (3). Because γm corresponds to the decay rate γ at q = 1, wr and wnr can be calculated from the relation q = wr/(wr + wnr) together with Eq. (3). A plot of the wr and wnr values is shown in the inset of Fig. 3(a). wr initially decreases with increasing λ and then increases for higher λ values. The order of wr is a few milliseconds, which is within the range of the theoretical radiative-rate values (sub-milliseconds to several tenth of milliseconds) reported in the literature (crosses) . However, the wavelength dependence of wr in the longer wavelength region is opposite to that reported in the literature. This may be due to phonon-assisted light emission occurring especially in smaller Si nanocrystals. However, wnr shows a monotonous decrease with increasing λ. This tendency is essentially the same as those observed in CdSe quantum dots  and Si nanocrystals in SiO2  and can probably be attributed to the suppressed density of surface nonradiative centers in larger nanocrystals .
Next, we discuss the reasons for the change in the width of decay rate distribution due to the deposition of Au films. As shown in Fig. 2(b), the width of the decay rate distribution increases with increasing Au thickness. Based on a statistical model of time-resolved decay curves used for an ensemble of emitters , it is understood that the broader is the decay rate distribution, the larger is the decay rate fluctuation of emitters. Thus, the broader distribution in ρ(w) obtained for the thicker Au layer samples [Fig. 2(b)] indicates that the decay rate fluctuation is higher in these samples. The decay rate is directly related to the LDOS and the broader distribution, ρ(w), corresponds to its spatially larger LDOS fluctuation (not temporally larger fluctuation).
To determine the fluctuation in the decay rate w, we calculated the standard deviation of ρ(w), which is defined by the square root of the variance of ρ(w). The variance of the decay rate, wvar, can now be obtained as follows from Eq. (1)Eqs. (9) and (10), σ is calculated from the fit-determined parameters wmf and wd. In Fig. 4, σ is plotted as a function of λ for the PSi and PSi/Au samples. The σ (PSi/Au) values of the PSi/Au samples are larger than those of the PSi sample σ (PSi) for all wavelengths. The larger σ (PSi/Au) values correspond to the larger decay rate fluctuations in these samples. Furthermore, the wavelength dependence of σ (PSi/Au) is clearly different from that of σ (PSi).
In the following, we discuss a possible origin for the larger fluctuation in the decay rate of the PSi/Au samples. Because the surface of the PSi layer has a root-mean-square roughness of the order of sub-nanometers to a few nanometers , the uncertainty of the distance between deposited Au nanoparticles and porous layers is considered to be of the same order. Such an uncertainty in the metal–emitter distance could lead to fluctuations in the LDOS. This is because the change in the LDOS due to the surface plasmon excitation is strongly dependent on this distance [1, 25]. In other words, the excitation rate of surface plasmons could be largely influenced by the uncertainty in the metal–emitter distance. Therefore, the larger-decay rate fluctuation [i.e., σ (PSi/Au) > σ (PSi) observed in Fig. 4] is attributed to the fluctuations in the surface-plasmon excitation rate.
To explain the experimentally observed large decay rate fluctuation in the PSi/Au samples, we calculated the standard deviation of decay rate based on the CPS model considering the uncertainty in the Au–PSi distance. Here, we define the width of decay rate distribution as , where γ(±) represent the decay rates calculated with SiO2 layer thicknesses of t ∓ δt, respectively [see Fig. 3(b)]. Introducing and wmf determined from the fit of the experimental decay curves into Eq. (9), the standard deviation in the presence of Au [σ(Au)] can be calculated using Eq. (10). The standard deviation of the PSi/Au samples [σ(PSi/Au)] is estimated under the assumption: σ(PSi/Au) =σ(Au)+σ(PSi). We used numerical parameters such as t, PSi layer thickness, and dielectric functions of Au layer, which were also used for calculating γ [Fig. 3(a)]. The uncertainty value δt was determined to be δt = 0.2 nm from the best fit with the experimental data.
In Fig. 4, the calculated results of σ(PSi/Au) are shown by solid curves. The calculated σ(PSi/Au) values show very good agreement with the experimental data (solid squares). This agreement suggests that the increased fluctuation of the decay rates can be attributed to the small uncertainty of the Au–PSi distance, and thus, to the small fluctuation in the excitation rate of localized surface plasmons.
As seen in Fig. 4, the wavelength dependence of σ(PSi/Au) is well reproduced by the present calculation. The obtained dependence mainly arises from that of σ (Au) given in the inset of Fig. 4. This fact indicates that the excitation rate of surface plasmons fluctuates more at longer emission wavelength, λ. Thus, the change in the surface-plasmon-induced LDOS becomes more pronounced at longer λ. Similar observations have also been made in semicontinuous silver films by using a near-field scanning microscope .
We investigated modifications of the emission decay rate of Si nanocrystals due to the deposition of thin semicontinuous Au films with different mass thicknesses over the wide emission wavelength range. The emission decay rate distribution was determined by fitting the experimental decay curves to the lognormal-distribution decay model. The mean and standard deviation values of decay rate distribution were successfully estimated and found to be modified owing to the deposition of Au films. These values were also found to depend on the emission wavelength and the thickness of Au films, and were well explained by a point-dipole decay rate model by introducing effective dielectric functions of the semicontinuous Au medium to describe the anomalous absorption peaks caused by the excitation of localized surface plasmons. The present approach affords a good understanding of the complex decay dynamics and can be applied to other luminescent systems comprising semicontinuous metal films such as nanoparticles arrays.
This work was partly supported by Grants-in-Aid for Young Scientists (B) (22760050) and Scientific Research (B) (23360133) from the Ministry of Education, Culture, Sports, Science and Technology, Japan, The Iketani Science and Technology Foundation, and The Murata Science Foundation. We are very grateful to Ms. N. Nabatova-Gabain and Dr. T. Moriyama (HORIBA Ltd.) for their support with the spectroscopic ellipsometry measurements and analysis.
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