Quenching the photoluminescence from Si nanocrystals of smaller sizes in dense ensembles due to migration processes

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Quenching the photoluminescence from Si nanocrystals of smaller sizes in dense ensembles due to migration processes V.A. Belyakov, K.V. Sidorenko, A.A. Konakov, A. V. Ershov, I.A. Chugrov, D.A. Grachev, D.A. Pavlov, A.I. Bobrov, V.A. Burdov

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S0022-2313(14)00330-5 http://dx.doi.org/10.1016/j.jlumin.2014.05.038 LUMIN12723

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Journal of Luminescence

Received date: 15 February 2014 Revised date: 6 May 2014 Accepted date: 28 May 2014 Cite this article as: V.A. Belyakov, K.V. Sidorenko, A.A. Konakov, A.V. Ershov, I. A. Chugrov, D.A. Grachev, D.A. Pavlov, A.I. Bobrov, V.A. Burdov, Quenching the photoluminescence from Si nanocrystals of smaller sizes in dense ensembles due to migration processes, Journal of Luminescence, http://dx.doi.org/10.1016/j. jlumin.2014.05.038 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Quenching the photoluminescence from Si nanocrystals of smaller sizes in dense ensembles due to migration processes V.A. Belyakov, K.V. Sidorenko, A.A. Konakov, A.V. Ershov, I.A. Chugrov, D.A. Grachev, D.A. Pavlov, A.I. Bobrov, V.A. Burdov† N.I. Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia 603950 We investigate both experimentally and theoretically the role of migration effects in a relaxation of the ensemble of Si crystallites formed in multilayer SiOx/SiO2 nano-periodic structure with annealing. Photoluminescence spectrum of the multilayer ensemble turns out to be strongly redshifted and narrowed related to its position and width expected from the size distribution of the nanocrystals. Based on the concept of “quantum confinement” we have performed a computer experiment on the ensemble relaxation with taking the migration of electrons, holes, and excitons into account for the ensemble having the size distribution and nanocrystals’ density similar to those in the experimental sample. This model allows us to calculate the photoluminescence spectrum that agrees well with the one observed experimentally. It was shown experimentally and theoretically that in a dense ensemble, the migration quenches the luminescence of Si nanocrystals whose diameters are, in fact, less than the average nanocrystal diameter in the ensemble. This explains the above mentioned features of the measured spectrum. Keywords: silicon nanocrystal; ensemble of nanocrystals; migration; photoluminescence; transition rate

1. Introduction At the beginning of the 1990th there was suggested an idea of possible “straightening” silicon band gap by nanostructuring [1-3]. At present, various methods of preparation (such as: ion implantation [4-7]; chemical vapor deposition from a gas phase [8]; magnetron sputtering [9,10]; colloidal synthesis [11]; electron beam epitaxy [12,13]; and some others) allow one to create nearly spherical 1 – 10 nm silicon crystallites capable of emitting light at room temperature. Nevertheless, light emission in Si crystallites is a result of strong concurrence with various nonradiative processes prohibiting (or, at least, influencing) the photoluminescence (PL). It is possible to partition these processes into intra-crystallite and inter-crystallite ones. Very often nanocrystals are considered as isolated. In this case only intra-crystallite processes take place, such as, e.g., radiative interband recombination, Auger recombination, and capture on surface defects. Meanwhile, in experiments, as a rule, one has to deal with large ensembles of nanocrystals. These ensembles are characterized by dispersion in nanocrystal sizes and nonradiative energy exchange between nanocrystals, which can essentially affect the ensemble PL. Such an energy transfer is occurred owing to a migration (inter-crystallite process) of elementary excitations––electrons, holes, or excitons. Migration of electrons and holes is of tunnel type, while excitons virtually migrate over the ensemble through the Förster-Dexter mechanism [14,15] due to the Coulomb interaction of excitons in adjacent nanocrystals. Previous experimental investigations of multilayer SiOx/SiO2 structures [16-18], porous Si [19], three-dimensional (3D) ensembles of Si [20,21] and A2B6 [22-25] crystallites demonstrated a dependence of their optical properties on the nanocrystal density and spatial arrangement, which                                                              †

Corresponding author. E-mail address: [email protected]

was treated as a migration’s manifestation. However, no quantitative estimations have been so far made for the contribution of the migration processes to the overall relaxation dynamics of an ensemble. In this paper, we report on our complex (theoretical, computational, and experimental) study of the PL and migration in an ensemble of Si nanocrystals. Here, we develop rigorous computational model describing the ensemble de-excitation and quantitatively explaining the experimentally observed PL spectrum. Our model confirms a determinative role of the tunnel migration in the relaxation of dense ensembles of Si nanocrystals, and reveals strong effect of the migration on the PL. We simulate the PL and migration in the multilayer (20 layers) composition of Si crystallites with ~ 105 crystallites in each layer taking into account the intra-layer and inter-layer energy transfer between the crystallites, as well as the radiative electron-hole recombination, capture on dangling bonds, and Auger recombination inside the crystallites. In this computer experiment, we used the size distribution of nanocrystals and volume filling factor similar to those taking place in the real multilayer nanoperiodic structure with which the experiment was carried out. In contrast to the suggested earlier qualitative and simplified computational model of light emission in a 3D ensemble of Si crystallites [26,27], we use here the directly computed rates of all the relaxation processes which allows us to obtain not only qualitative but also quantitative agreement with experimentally measured PL spectrum of the multilayer composition. This is, presumably, also due to the large number of nanocrystals involved into the carried out here computer experiment related to the computer model developed in Refs 26, 27 (no more than 5000 nanocrystals). As a result, it is possible to obtain a correct statistics for emitting nanocrystals in our case. Our theoretical study is based on the concept of the so-called “quantum confinement”. We suppose here that light emitted by the considered multilayer nanoperiodic composition originates exclusively from radiative interband transitions between size-quantized electronic states localized inside the nanocrystals. Similarly, all the nonradiative processes are considered as occurring within the nanocrystals but not inside the host matrix. Such an approach, as was already mentioned, makes it possible to explain experimentally observed PL spectrum of the nanoperiodic structure.

2. Experimental In the experiments we employ a-SiOx/SiO2 multilayer nano-periodic structure with x ≈ 1.5 deposited by alternate vacuum evaporation of the corresponding initial materials from two separate sources as described in Ref. 28. The a-SiOx layers were formed by the evaporation of a chemically pure SiO dust from an effusion tantalum source, and the a-SiO2 layers were formed by the electron-beam evaporation of fused silica. During deposition, the substrate temperature was kept at 200 ± 10°C. The substrates were p-Si:B (100) wafers with the resistivities 12 Ω·cm. Before deposition, the residual atmospheric pressure was 1.5×10–6 Torr. During deposition, the SiO and SiO2 vapor pressures were 7×10–6 and 2×10–4 Torr, respectively. The thicknesses of the a-SiOx and SiO2 layers were set to 4 nm and 3 nm, respectively. The first layer deposited onto the substrate was a SiOx layer; next we deposited a SiO2 layer, then an a-SiOx layer, etc. The upper layer was a SiO2 one. After deposition of the layers, the samples were subjected to annealing in a nitrogen atmosphere at 1100°C during two hours. The layers and the nanocrystals formed after annealing are shown in Fig. 1. The growth of the nanocrystals is almost two-dimensional because the nanocrystals form, in fact, in SiOx layers. Such an approach allows one to control better the sizes of the grown nanocrystals because their average size is close, as a rule (see, e.g., Refs 10, 29), to the widths of the initially fabricated aSiOx layers. High-resolution transmission electron microscopy (HRTEM) was applied to study

the cross-sections of the samples. These measurements were carried out on the microscope JEM2100F (JEOL) operating at 200 kV. Left image in Fig. 1 was obtained in the dark field scanning mode of the transmission electron microscope (STEM), while the central image in Fig. 1 was recorded in HRTEM mode. The samples were prepared using standard technique with equipment of the Gatan company.

Fig. 1. The cross-section STEM (left panel) and HRTEM (central panel) images of the multilayer nanoperiodic structure nc-Si/SiO2 after annealing at 1100°С. Fourier transform of the HRTEM image is presented on the right.

In the central image of Fig. 1 one can see some clusters with greater sizes and complex structure looking strongly imperfect and having no entire crystal lattice. In order to understand the structure of these objects we analyze them using the filtered Fourier transforms in the real and reciprocal spaces, as shown in Fig. 2. We choose one of such objects with linear size greater than 10 nm, the HRTEM image of which is shown in Fig. 2 (a). As seen, this object consists of several fractions with different orientations of the crystal lattice providing typical diffraction pattern with individual point reflexes in the k-space (result of the Fourier transform) (Fig. 2 (b)). We choose two point reflexes labeled as “C1” and “C2” and make the inverse Fourier transform of these two point reflexes with some their vicinities. The result is shown in panel (c). It is clearly seen that the inverse Fourier transform yields the images which can be interpreted as two separate nanocrystals with the sizes about 3 nm (“C2”) and 4 nm (“C1”). These two nanocrystals correspond to two crystalline fractions marked in Fig. 2 (a) with white circles. Evidently, all the other point reflexes and their nearest surroundings in panel (b) will create similar images of nanocrystals after the filtered Fourier transform. It is possible to suppose that the great multi-fractioned objects in HRTEM images, similar to the one shown in Fig. 2 (a), represent several nanocrystals laying one over another, whose images are superimposed one on another.

Fig. 2. Fragment of the HRTEM image of the ensemble of Si nanocrystals (a), and its Fourier transform (b) representing some “diffraction pattern” with point reflexes. Inverse filtered Fourier transform of the reflexes labeled by “C1” and “C2” yields two separated real-space images shown in panel (c), which correspond to two crystalline fractions indicated in the panel (a) with white circles. Arrow in panel (a) indicates the nanocrystal of extremely small size (less than 2 nm).

Presumably, formation of amorphous clusters, predominantly with small sizes, is as well possible. However, chosen low contrast of the HRTEM images does not, in fact, allow us to single out the amorphous Si clusters against the amorphous matrix background. On the contrary, the nanocrystals look in HRTEM like dark spots with some regular structure. Therefore, vast majority of the observed with HRTEM nanoclusters have a crystalline structure. Only these nanoclusters, which are nanocrystals, are taken into account in the subsequent simulations. Processing the TEM data for chosen multilayer composition allows one to determine the size distribution (shown in Fig. 3 by vertical columns with dots) of the nanocrystals formed after annealing in the a-SiOx layer. It was found that the distribution over nanocrystals’ radii within the range 0.5 nm < R < 5 nm is well fitted by the function (dashed line in Fig. 3): 2 ρ ( R ) = 0.45N ( R − 0.25) exp ⎡ −0.64 ( R − 1.2 ) ⎤ , (1) ⎣ ⎦ where the radius is taken in nanometers, and N stands for the total number of nanocrystals in the ensemble. According to our analysis of the TEM images, no nanocrystals with the radii greater than 5 nm, and smaller than 0.5 nm exist in the ensemble. As seen from Fig. 3, small nanocrystals are resolved with lower accuracy mainly due to the greater relative deviations of their shape from a spherical one. This results in a higher uncertainty in their size distribution as shown in the figure with greater error bars. Nevertheless, TEM-observation of the nanocrystals even with R < 1 nm is, in principle, possible. For example, in Fig. 2 (a) one can see such a small nanocrystal indicated by an arrow. In accordance with Eq. (1), mean value R and standard deviation δ R for the explored system are: R = 1.88 nm; and δ R = 0.69 nm.

Fig. 3. Size distribution of the nanocrystals in the ensemble. Dots: relative fraction of the nanocrystals with corresponding sizes resulting from the HRTEM data. Dashed line: fitting the HRTEM size distribution by the function (1). Solid line: result of recalculation of the experimental PL spectrum into the size-distribution function according to Eq. (2).

The examined multilayer structure was excited by pulsed N2-laser ( λex = 337 nm), and then it relaxed partially through the radiative channel. The pulse repetition frequency was ~26 Hz, the pulse duration was ~10 ns, and the excitation pulse energy was ~30 μJ. The PL signals were detected by photoelectric multiplier and SP-150 (Stanford Research Systems) grating monochromator with zero time-delay with respect to the laser pulse. The duration of detecting (gate) was equal to 0.1 s. The PL spectrum was corrected with respect to the spectral response of our measuring system and was recorded in the range of wavelengths 400 nm < λ < 900 nm as shown in Fig. 4 with solid line. It is possible to recalculate the size distribution of the emitting crystallites in the ensemble from the obtained PL spectrum using the following relationship between the PL intensity and the density of the size distribution ρ R ( R ) of the emitting crystallites:

ΔI PL = ω ( R ) ρ R ( R ) ΔR ,

(2)

where ω( R) is the radius-dependent frequency of the emitted photon, and ρ R ( R ) ΔR the amount of the emitting crystallites with radii ranging from R − ΔR 2 to R + ΔR 2 . Emphasize that ρ R ( R ) , generally speaking, should not coincide with ρ ( R ) defining by Eq. (1), since only a part of the ensemble crystallites participates in the photon emission. Indeed, recalculation according to Eq.(2) yields ρ R ( R ) which essentially differs from the one obtained from the TEM data, as shown in Fig. 3 with solid line. As seen, the distribution function ρ R ( R ) becomes less than ρ ( R ) beginning with the nanocrystal sizes close to the mean size in the ensemble, and reduces with further decreasing the nanocrystal size. Evidently, nanocrystals with greater sizes mainly participate in the light emission, while the smaller nanocrystals relax nonradiatively. It is possible to suppose that small nanocrystls transfer the energy of excitation into the greater ones, where the radiative interband transitions occur.

3. Computational details In order to corroborate this hypothesis, we numerically modeled the ensemble PL and migration. As a part of modeling the plane-parallel layers containing 105 Si nanocrystals in each layer were generated with the size-distribution described by Eq. (1), and the volume filling fraction typical for the experimental sample. In particular, the investigated sample had the filling factor not less than 30%. The same value was used in our computational model. Randomly chosen 50 % of the nanocrystals had a dangling bond at their surface. Initially, all the nanocrystals are assumed to be equivalently excited: each of them at t = 0 has three electron-hole pairs (excitons) inside. In what follows, these electron-hole pairs relax through various (both radiative and nonradiative) channels. Simulating the PL we consider five competitive processes for each nanocrystal: (i) radiative interband recombination (rate τ R−1 ); (ii) capture of the electron or hole on a dangling bond (rate

τ C−1 ); (iii) Auger recombination (rate τ A−1 ); (iv) tunneling of the electron or hole between nanocrystals (rate τ T−1 ); (v) Förster exciton transition from one nanocrystal to some other (rate

τ F−1 ). One more event can be realized per unit time apart from the five mentioned above. This is the so-called “zero” event meaning that the system remains in its given state. Each of the relaxation or migration processes has the probability equal to a product of the rate and the timestep δ t . The probability P0 of the “zero” event is defined by the normalization condition:

(

)

δ t τ R−1 +τ C−1 + τ A−1 + τ T−1 +τ F−1 + P0 = 1 .

(3)

δ t was so chosen that maximal value of δ t τ for the fastest process among all the nanocrystals in the ensemble be less than 0.01. Moreover, total amount of the events during the time-step in the ensemble was limited by 1000. The choice of the event occurring in each nanocrystal for the time δ t is made with the Monte-Carlo procedure by putting a random number on the probability scale. The rates of all the considered processes were calculated earlier using the Fermi’s golden rule. It was found that τ A−1 varies within 106 – 1012 s-1 [30-33], while τ C−1 varies from 102 to 1010 s-1

[34]. The radiative electron-hole recombination is essentially slower process. Its rate τ R−1 is always less than 106 s-1 [26,35-37]. Tunneling of excited carriers and exciton transitions are sensitive to both crystallite sizes and inter-crystallite distances. When the nanocrystals touch one another, the tunneling rates τ T−1 are high enough: 107 – 1012 s-1 depending on the relationship between the nanocrystal sizes. As the inter-crystallite distance increases up to 1 nm, the rate exponentially drops, and becomes equal to 102 – 108 s-1 [38,39]. At last, the slowest process among all the considered––Förster exciton transfer––has the rates τ F−1 which do not exceed 104 – 105 s-1 [39,40]. However, these calculations were carried out, as a rule, for crystallites with radii ranging from 1 to ~ 3 nm. This interval covers an essential part of the crystallites in the modeled ensemble but not their total amount. There are crystallites in the ensemble with smaller or greater sizes as follows from Fig. 3. On this reason, in the following, we shall use for the rates some approximate expressions obtained by fitting to the correct calculated dependencies [26,30-40], and extrapolated into R < 1 nm, and R > 3 nm domains. Note that, in all the approximate formulae the crystallite radius, as well as the distance between the crystallites (for the migration processes), are given in nanometers. Correspondingly, the rates will be computed in inverse seconds. Below we adduce these approximate analytical expressions exploited in our computational model. The rates are defined at room temperature by the following expressions: 105 1.12 + 1.95 R1.8 , (4) τ R−1 = 3 3.07 R

(

)

τ A−1 = 5 × 1015 exp −13.8R 0.36 ,

(5)

⎧⎪ 24 ( R − 0.67 )2 Θ ( R − 0.67 ) ⎫⎪ ⎧⎪ 24 ( R − 0.74 )2 Θ ( R − 0.74 ) ⎫⎪ −1 , exp = τ ⎬ ⎨ ⎬, C (h) 2.05 2.07 0.1 R 0.67 0.12 R 0.74 + − + − ( ) ( ) ⎩⎪ ⎭⎪ ⎩⎪ ⎭⎪

τ C−1( e ) = exp ⎨

4.3 × 109 R −5.6b −2 exp ( −9.4 L )

τ T−1( e ) = 10

−4

+ ⎡0.9 ⎣

(

R1−1.8

−

R2−1.8

) − 0.05⎦⎤

, τ T−1( h ) = 2

1010 R −5.6b −2 exp ( −10.7 L ) 10

−4

+ ⎡1.05 ⎣

(

R1−1.8

− R2−1.8

) − 0.07⎦⎤

2

(6) , (7)

1380 b6

τ F−1 =

, (8) 2 0.0005 + ⎡1.95 R1−1.8 − R2−1.8 − 0.1075⎤ ⎣ ⎦ where Θ ( x ) is the step-function of argument x . Here, we do not distinguish the rates of eeh and

(

)

ehh Auger processes because of their relatively small difference. On the contrary, capture on surface defects has substantially different rates for the electrons and holes, especially in small nanocrystals. b is the center-to-center interdot distance equal to L + R1 + R2 = L + 2 R . Note also that when considering the migration processes, we imply the transitions of excitations occurring from smaller nanocrystals to greater ones. In the opposite case, the excitation has to increase its energy by Δε that is the difference of the excitation energies in the first and the second nanocrystals. Accordingly, in this case, the probability of the transition is weakened by the factor exp ( − Δε k BT ) [41]. Finally, we also use the following approximation for the optical gap dependence on the nanocrystal radius [42]:

ε g ( R ) = 1.12 +

1.95

, R1.8 where 1.12 eV is a band-gap of bulk Si at room temperature.

(9)

It is worth noting also that we neglect in the subsequent simulations the Coulomb blockade effect which influences the electron and hole tunneling. We, first, assume that the resonant tunneling will be infrequent, and a main part of the tunnel transitions occurs non-resonantly. This means that phonon (or, even, multi-phonon) assistance is required for the transition. Second, we assume that the difference between the energy levels in neighboring nanocrystals is great enough compared to some typical Coulomb energy because typical size-quantization energies in nanocrystals are substantially greater than the Coulomb energies. Such an assumption is justified for nanocrystals whose sizes do not exceed several nanometers. Precisely those nanocrystals are considered here.

4. Results and discussion Results of our modeling are depicted in Fig. 4 by dashed and thick solid lines. Experimental spectrum is shown by thin solid line. In fact, we calculate and measure the total energy (the number of photons) emitted by the ensemble during 0.1 s after the laser pulse (i.e. the PL intensity integrated over time) at different wavelengths. In the computer experiment two different cases were modeled. In the first case, we “switch off” all nonradiative processes including the migration (dashed line in the figure). Correspondingly, all the initially excited nanocrystals emit photons. In this case, the total number of emitted photons equals the triple number of the nanocrystals in 20 layers: 3×20×105 = 6×106; because three electron-hole pairs are excited in each nanocrystal. In the other case, all the nonradiative processes are “switched on” and compete with the radiative recombination. The modeled spectrum is depicted with thick solid line. The calculated quantum efficiency of the photon generation turns out to be small enough and equal to 1.22 %, i.e. the total number of emitted photons approximately equals 7.3×104. In Fig. 4 we also show with an arrow the wavelength (708 nm) of the photon emitted by the nanocrystal with R = 1.88 nm.

Fig. 4. Normalized and integrated (over time) PL spectra: experimental, obtained after N2-laser (λ = 337 nm) excitation (thin solid line); modeled, with taking the migration and other nonradiative relaxation processes into account (thick solid line); modeled, with only the radiative recombination to be allowed (dashed line). Double-headed arrow indicates the photon wavelength corresponding to the meanradius nanocrystal.

As seen from the figure, the modeled PL spectrum at λ > 600 nm with high accuracy agrees with the observed one, if the migration is taken into account. At the same time, if we neglect the migration, the modeled dependence I PL ( λ ) strongly differs from the experimental data. This corroborates the hypothesis about the energy transfer from the smaller nanocrystals to the greater ones where the energy releases through photon generation. One more indication on the migration in the ensemble is a stretched exponential decay in a µs time-scale, which is usually treated as an overall contribution to the PL from equivalent (in size) nanocrystals having different PL lifetimes [16,43,44]. We have extracted the time evolution of the PL signal from our computer

experiment for the photon-energy window 1.45eV < ω < 1.5eV , which approximately corresponds 825nm < λ < 850nm , and 2.5nm < R < 2.7 nm according to Eq. (9).

Fig. 5. Microsecond decay of the PL at 825nm < λ < 850 nm extracted from the computer experiment (dots), and fitted by two different exponents (solid lines) on the initial and final stages as well as by stretched exponent (dashed line) with parameters indicated in the figure.

The results are shown in Fig. 5. Fitting the extracted dependence by the stretched exponent β exp ⎡ − ( t τ ) ⎤ yields τ ∼ 5 µs and β ∼ 0.45 . As seen, such a fitting is not quite satisfactory. ⎣ ⎦ More realistic description of the time evolution of the PL signal in our model can be based on a multi-exponential form of the PL intensity [45]. In this case, it is possible to reveal two single exponents with different decay time: 5 µs; and 400 µs as shown in the figure. The faster component is presumably responsible for the finishing migration (t 10 µs), while the slower one describes the radiative recombination (t 30 µs). In the transient region both processes have comparable intensities. As follows from the results of both real and computer experiments (Fig. 4), the migration of excitations leads to two important effects. First, the PL peak becomes essentially narrower compared to the case of isolated nanocrystals. Second, appreciable redshift of the peak takes place. Thus, the performed computational simulation confirms an important role of the migration processes in the relaxation dynamics of ensembles of Si crystallites when the crystallites are in close proximity. Also, as seen in Fig. 4, weak high-energy PL band with a peak centered at ~ 530 nm was detected in experiments. According to the figure, the description of this PL band in the frames of our computational model is, in fact, unsatisfactory. Our simulations do not exhibit even weak PL in this range. Frequently, this PL band is associated with different possible oxygen centers [4649]. In our model, however, we do not consider such defects. Therefore, there are no reasons to expect light emission in this range within the frames of our computational model. One more reason for the absence of the PL peak at 530 nm in the simulations can be caused by a certain inaccuracy in determining the capture rates at small R, where these rates sharply vary by several orders of magnitude. It is interesting to trace time evolution of the emission spectra of the considered system in the framework of the suggested computational model. In Fig. 6 we show the electromagnetic energy (integrated over time PL intensity) emitted by the ensemble during the first 10-6 s, 10-5 s, 10-4 s, 10-3 s, and 10-2 s vs photon wavelength. As seen, the main part of the electromagnetic energy is radiated between 10-4 s and 10-3 s. Indeed, precisely these values are typical for the radiative lifetimes of nanocrystals with 1.5nm < R < 3nm , which represent, in turn, the main part of the emitting nanocrystals in the ensemble according to Fig. 3 (solid line). At the same time, after 10-

2

s, the ensemble does not, in fact, emit, and the integrated over time PL spectrum profile remains almost invariable. Therefore, the difference in wavelength dependencies of the emitted energy after 10-2 s and after the full time 0.1 s (shown in Fig. 4 by thick solid line) is negligibly small. It is also seen that the PL peak is slightly redshifted with time. This is a manifestation of finishing the migration processes. Evidently, they have been mainly finished at 10-6 – 10-5 s.

Fig. 6. PL spectra after the emission time indicated in the figure. The PL spectrum after 10-2 s almost coincides with the one obtained after the full time 0.1 s.

Mention, once again, that our treatment is based exclusively on the “quantum confinement” model which implies that the PL is caused by the transitions between the quantum states confined inside the nanocrystals. Such a point of view is confirmed experimentally for Si nanocrystals embedded in different matrices [50,51]. Nevertheless, for small Si nanocrystals whose sizes do not exceed 1.5 – 2 nm in diameter the mechanism of light emission is still under extensive debate. There are different points of view [52] depending on the kind of the nanocrystal surface. It has been shown, in particular, that the presence of Si–O–Si bond at the oxidized nanocrystal surface [53] results in the localization of the electron wave function on this bridge bond, and reduction of the nanocrystal optical gap that is the energy of the surface-type HOMO-LUMO transition in this case. It has been also revealed [54,55] that small nanocrystals embedded in an a-SiO2 matrix tend to be amorphous and strained. The wave function in those crystallites spreads over the crystallite volume and interface region. The amorphization and strain leads as well to a remarkable decrease of the HOMO-LUMO gap [54,55] and suppression of the radiative transitions. We have to point out, however, that amorphous clusters are not, in fact, seen in our TEM images, and the size distribution presented in Fig. 3 was built exclusively for the nanocrystals. Thus the amorphous clusters, even in case of their presence in the ensemble, were not taken into account in our simulations, and were considered, in essence, as dark. In any case, according to Fig. 3, the  fraction of the nanocrystals whose radii are less than 1 nm is small enough. Therefore, it is possible to suppose that the contribution of smallest nanocrystals to our computer experiment has no a crucial value. Meanwhile, the main part of the nanocrystals in the ensemble has the radii ranging within 1 – 3 nm, for which the “quantum confinement” model is quite justified [50]. In a certain sense alternative view on the mechanism of the PL of silicon nanocrystals embedded in silicon-dioxide matrix was proposed in Refs 44, 56. The authors give some experimental confirmations of that the PL is caused by some emitting centers located in a-SiO2, such as doubly bonded Si = O defects, but not by nanocrystals, which remain “dark”. At the same time, the authors do not explain why in this case the PL spectra are usually strongly broadened and demonstrate the PL-peak shifting as the nanocrystal sizes change. Moreover, if the PL is not originated from nanocrystals, then, presumably, their presence is not at all necessary in a-SiO2 matrix. Nonetheless, amorphous silicon dioxide itself does not exhibit the PL typical for nanocrystalline silicon dioxide. Therefore, it seems that the question about the emitting Si = O centers in a-SiO2 remains so far open.

5. Conclusion Thus, in our study we have demonstrated that migration of excited carriers in closely packed ensembles of Si nanocrystals crucially influences the ensemble PL. In particular, the migration leads to being out the small nanocrystals of light emission. This causes the red-shifting and narrowing the PL peak of the ensemble. It seems also relevant to indicate the following circumstance concerning experimental determining the optical gap of Si nanocrystal as a function of its radius: ε g ( R ) . In experiments, the PL peak position is attributed, as a rule, to the wavelength (or the energy) of the photon emitted by the nanocrystal with mean size in the ensemble. In our case this wavelength equals 708 nm, as shown in Fig. 4 by the arrow. As seen from the figure, this value turns out to be less by ~ 100 nm than the experimentally observed peak position. Correspondingly, in order to find experimentally the correct dependence ε g ( R ) , one has to measure the mean size of the emitting nanocrystals only but not of all nanocrystals in the ensemble. It is clear, that such a measurement is, in fact, impossible. Therefore, very narrow size distribution of nanocrystals is desirable to minimize the mean-size difference for the whole ensemble and its emitting part.

Acknowledgements We thank prof. I.N. Yassievich and prof. O.B. Gusev for their interest to this work and helpful discussions. The work was supported by the Russian Foundation for Basic Research (grants No 14-02-31637, No 14-07-00582, and No 14-02-00119), the Russian Ministry of Education and Science (Federal Task 2014/134, project No 2696), and the “Dynasty” foundation.

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Highlights:    1. Simulation of luminescence in dense ensembles of silicon nanocrystals was carried out    2. Concurrence of radiative transitions and energy transfer is studied in the ensemble    3. Smaller nanocrystals transfer their excitation to the greater ones, and become “dark”