IOP PUBLISHING
NANOTECHNOLOGY
Nanotechnology 20 (2009) 135702 (8pp)
doi:10.1088/0957-4484/20/13/135702
Influence of SiO2 matrix on electronic and optical properties of Si nanocrystals K Seino1 , F Bechstedt1 and P Kroll2 1
Institut f¨ur Festk¨orpertheorie und -optik, Friedrich-Schiller-Universit¨at and European Theoretical Spectroscopy Facility (ETSF), Max-Wien-Platz 1, 07743 Jena, Germany 2 Department of Chemistry and Biochemistry, The University of Texas at Arlington, Arlington, TX 76019, USA E-mail:
[email protected]
Received 5 August 2008, in final form 9 January 2009 Published 11 March 2009 Online at stacks.iop.org/Nano/20/135702 Abstract Based on ab initio density functional theory we present electronic properties and the optical response for Si nanocrystals embedded in amorphous SiO2 networks. Quasi-spherical dots with diameters from 0.8 to 1.6 nm are investigated. The results for Si nanocrystals embedded in SiO2 are compared with corresponding results for hydrogenated Si nanocrystals of the same size. The calculations show the influence of the interface between nanocrystal and matrix on the electronic properties. The results are compared with recent experimental data and discussed in detail. As striking features, strong reductions of the gaps and their diameter variation are predicted due to the oxide presence. Electronic confinement mainly influences the absorption edge while at higher photon energies only broad peaks at almost fixed positions occur. (Some figures in this article are in colour only in the electronic version)
advantage of low cost, abundant, non-toxic, stable, and durable materials. Nanoscale structures, e.g. multi-quantum well structures and embedded nanocrystals, allow the fabrication of higher band-gap materials that can be combined in tandem solar cells with bulk Si cells [2]. The quantum confinement not only allows one to tailor the effective band-gap but also to improve the performance of hot-carrier solar cells. Recently, for the first time multiple exciton generation in Si NCs was reported [3]. For Si NCs embedded in SiO2 many experiments have been carried out. In particular, the confinement effect and its size dependence have widely been studied [4–8]. Measurements of optical properties have also been reported for embedded Si NCs [9–11]. In contrast, theoretical studies were mainly focused on hydrogen-passivated free-standing Si nanodots because of the computational challenges due to the amorphous matrix. Hydrogenated silicon clusters have been studied theoretically by many groups over the last decade [12–16]. The energy gaps, excitation energies and optical properties have been discussed and many-body effects such as self-energy corrections and excitonic effects have sometimes been taken into account [12, 13]. The central goal was to study the influence of quantum confinement which increases the energy of the optical and quasiparticle gaps in silicon clusters.
1. Introduction In recent years there has been considerable interest in nanostructured silicon since it is a promising material for quantum devices of the next generation. Effects due to spatial quantization help to overcome the limitation of the indirect-gap semiconductor Si for light emission. It has been found that the band-gap of Si nanocrystals (NCs) increases with decreasing NC size and is accompanied by an efficient luminescence. Interesting Si-based nanostructures are Si NCs embedded in an insulating matrix, such as amorphous SiO2 or Al2 O3 . New promising approaches for the fabrication of sizecontrolled Si NCs in SiO2 have been proposed [1]. Using size control, the difficulty of a quantitative analysis of the photoluminescence (PL) signal due to the broad size distribution of the Si NCs could be at least partially solved. It leads to a better performance and opens up a way to active optoelectronics applications. Si NCs embedded in a SiO2 matrix represent a promising material for Si-based optoelectronic devices such as light-emitting diodes, light detectors or solar cells. The photovoltaic application of Si nanostructures is driven by the predicted enormous increase of the light conversion efficiency [2]. Si- and SiO2 -based systems also have the 0957-4484/09/135702+08$30.00
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Theoretical studies directed to the role of oxygen in Si NCs were performed for isolated, partially oxidized Si clusters with one or several surface O atoms [17–21]. The effect of surface passivation of Si NCs with atomic oxygen has been discussed. There are only a few calculations on particles with complete oxide shells. Zhou et al [22] studied the electronic structures of Si NCs with one oxide overlayer. In their models H atoms have been replaced by hydroxyl (–OH) groups or bridge O atoms at the cluster surface. Oxidized NCs with several oxide shells were considered by Ramos et al [23–25]. The effect of the progressive backbond oxidation and of the passivation with –OH groups was demonstrated. However, theoretical studies of confinement effects and optical properties for Si NCs embedded in SiO2 are still computationally challenging. The few theoretical studies of Si NCs embedded in SiO2 assume rather unrealistic situations or are restricted to classical simulations of the geometry. Daldosso et al [26] and Luppi et al [20] have performed density functional theory (DFT) calculations of structural, electronic and optical properties for a very small model with 10 Si atoms in the core (NC diameter d ≈ 0.55 nm). The β cristobalite structure was used as host SiO2 material and Si–Si distances have been assumed to be equal to those of Si–O– Si bonds, which causes unrealistic strained bonds. Atomistic simulations using a Monte Carlo or a molecular dynamics approach [27–29] usually neglect the quantum-mechanical effects on the bonding in the nanostructured systems. Only Kroll and Schulte [30] have studied geometries and electronic properties of Si NCs embedded in amorphous SiO2 (0.8 nm d 1.6 nm) using a fully quantum-mechanical DFT treatment. However, the influence of the matrix region and its interplay with the Si core are still open questions. The electronic structure of embedded dots, especially their optical gaps, and the consequences for the optical properties need an understanding in terms of size, interface bonding, and matrix contribution. In the present paper, we pursue the influence of a SiO2 matrix on embedded Si nanocrystals by using realistic models and first-principles calculations. In order to understand the matrix influence in detail, results for the electronic and optical properties of Si NCs of equal geometry with a SiO2 matrix and only a H-passivated surface are compared.
in [30]. At first, we construct a supercell of diamond silicon. We then define all Si atoms within a radius around a central Si atom as belonging to the inner core region; all remaining Si atoms belong to the outer matrix region. The diameter of the Si cores d varies in the range of 0.8, 1.0, 1.2, 1.4 and 1.6 nm. 3 × 3 × 3 Si diamond supercells are used for the starting geometries of the smaller Si NCs with diameters of up to 1.2 nm. For larger Si cores with diameters of 1.4 and 1.6 nm we apply much larger 4 × 4 × 4 supercells. The large edge length is required in order to avoid the existence of defect states [30] and the interaction of a nanocrystal with its images in the adjacent cells. The resulting quasi-spherical nanocrystals consist of 17, 29, 47, 71 and 99 Si atoms in the core, respectively. In the next step, we insert O atoms between pairs of Si atoms outside the core. By construction, only Si–O–Si bridge bonds practically form the interface, but all suboxide atoms Si1+ , Si2+ and Si3+ occur in the interface region in agreement with experimental observations [32]. After we define the central and outer regions, the structures are optimized at first using an empirical network method in order to decide on a local minimum for the geometry. The procedure is finished by additional optimization due to minimization of the DFT total energy. After geometry relaxation the nanocrystals exhibit smooth and defect-free interfaces to the amorphous SiO2 matrix. The hydrogenated Si NCs are constructed from the cores of Si atoms obtained from the Si NCs embedded in SiO2 , by removal of the SiO2 regions. The Si dangling bonds at the surface are passivated with hydrogen atoms. For the hydrogenated models, in addition, we allow the atomic geometry to relax around the passivating hydrogen atoms until ˚ −1 . the Hellmann–Feynman forces are smaller than 10 meV A Atomic geometries of the embedded and hydrogenated NCs obtained within this procedure are shown in figure 1 for the Si NC model with a diameter of 1.6 nm. 2.3. Pair excitation energies The optical excitations are modified by the electron–electron interaction. In general, they cannot be identified with differences of the Kohn–Sham (KS) eigenvalues of the DFT. However, in systems with strong quantum confinement the pair excitation energies can be calculated by means of the delta-self-consistent-field (SCF) method combined with an occupation constraint [14, 15, 23]. Such pair excitation energies, in which the Coulomb interaction of quasielectron and quasihole is included, can be determined as the difference in the total energies of the system in the ground state and the first electronically excited state. The pair excitation energy of a neutral pair follows as
2. Computational details 2.1. General The calculations are performed using the Vienna ab initio simulation package (VASP) implementation [31] of the density functional theory within the local-density approximation (DFT-LDA). The electron–ion interaction is described within the projector augmented-wave (PAW) method. The supercell approach is applied in order to use a plane-wave expansion of the eigenfunctions. The cutoff energy of plane-wave expansion is taken to be 400 eV. The Brillouin-zone (BZ) integrations are carried out using only the point.
E gex = E(N, e + h) − E(N)
(1)
with the total energy E(N) of the N -electron system in the ground state and the total energy E(N, e + h) of the N -electron system excited by an electron (e)–hole (h) pair. The energy E(N, e + h) is calculated with the occupation constraint that the highest occupied molecular orbital (HOMO) of the ground state contains a hole, while the lowest unoccupied molecular orbital (LUMO) contains the
2.2. Structure modeling The construction of optimized geometries for the Si NCs embedded in amorphous SiO2 has been described in detail 2
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Figure 1. Models of a Si nanocrystal with a diameter of 1.6 nm embedded in SiO2 (left) and a hydrogenated Si nanocrystal with same size (right). The 99 Si atoms of the core are indicated by green (large) balls, while Si (O, H) atoms in the matrix are indicated by yellow/middle (red/small, light blue/small) balls.
HOMO and LUMO levels, but also energetically more distant electronic levels, are plotted for Si NCs embedded in SiO2 and the corresponding hydrogenated NCs using Kohn–Sham eigenvalues from DFT–LDA. Our results for hydrogenated NCs are in good agreement with those of our previous works [14, 23] using slightly different structures and numerical treatments. The small differences are consequences of the slightly modified NC shapes and numerical treatments. The quantum-confinement effects on the energy levels are clearly visible for both types of NCs. The HOMO– LUMO gaps decrease with increasing diameter. This is in general agreement with experimental observations [8], that with large diameters the confinement influence vanishes and the gap values converge towards the bulk Si value, here E gbulk = 0.44 eV obtained within the DFT–LDA. However, for the largest studied embedded Si NC, there is still a large bandgap opening relative to bulk Si. Figure 2 also illustrates the significant influence of the matrix material on the electronic confinement. The energy differences between occupied and empty levels are drastically reduced in the presence of the SiO2 matrix. The matrix-induced gap reduction is larger than 30% for the smallest NCs, but becomes smaller than 20% for the two biggest NCs studied. The reduction of the quantum confinement is a consequence of the lowering of the energy barriers at the Si–SiO2 interface in comparison to the Si–H surfaces. Large differences between PL energies have also been observed experimentally by studying the emission of Si nanostructures before and after oxidation [38]. Figure 3 reports the calculated values of the HOMO– LUMO energy gaps E g and the pair excitation energies E gex (including quasiparticle and excitonic effects) for a variety of Si NCs with different sizes and different environments, Si NCs embedded in SiO2 and hydrogenated Si NCs. For the hydrogenated Si nanocrystals the calculated energies are similar to the results by Weissker et al [14] and Ramos et al [23] for NCs with almost the same number of atoms and a Si atom in the NC center. The values in figure 3 clearly demonstrate the confinement of the electrons and holes versus the diameter of the Si NCs. For hydrogenated Si NCs we confirm the result of an almost linear variation of
corresponding remaining electron. The pair excitation energies E gex rule the energetical positions of the emission peak in the PL or the onset of the optical absorption. In the PL case the energies are additionally influenced by the Stokes shift, which may be calculated by optimizing the system geometry in the presence of an excited electron–hole pair [33]. For the purpose of comparison, we also discuss single-particle gaps, i.e. HOMO–LUMO energy gaps, E g , as the difference of the Kohn–Sham eigenvalues of the LUMO and HOMO derived directly from the DFT-LDA. 2.4. Optical properties The optical properties are calculated within the independentparticle approximation [34], i.e. neglecting quasiparticle, excitonic and local-field effects. These effects are in general very important. However, for the energetical positions, especially the lowest pair energies, there is almost compensation of these effects and the DFT–LDA gives reasonable values [33]. The line shape is still influenced by the many-body effects [35, 36]. However, here we are more interested in the influence of an amorphous SiO2 matrix and its interplay with the Si NC core than in a detailed analysis of many-body interactions. The comparison of dielectric functions for hydrogenated and SiO2 -embedded Si NCs obtained within the independent-particle approximation should give the most important information. The optical transition matrix elements are calculated using all-electron wavefunctions obtained within the PAW method. This approach is not only applicable to bulk materials [37] but also to NCs [14, 15, 23–25]. A spectral broadening of 0.10 eV is used.
3. Results and discussions 3.1. Electronic properties It is well known, that the absorption edge of the NCs is characterized by transitions between the HOMO and LUMO states. In figure 2, the energy level diagrams including the 3
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Figure 2. Energy level scheme of (a) Si nanocrystals embedded in SiO2 and (b) hydrogenated Si nanocrystals with diameters from 0.8 to 1.6 nm. The HOMO levels define the energy zero. The values of the HOMO–LUMO gaps are marked by the arrows. DFT-LDA eigenvalues are used.
the gaps and pair excitation energies with the inverse NC diameter d , E g ≈ 0.44 eV + 2.50 eV/(d(nm))0.76 and E gex ≈ 0.44 eV + 2.60 eV/(d(nm))0.77 . The diameter variations are slightly weaker than those derived from other ab initio calculations [14]. The main reason may be the fact that previously investigated Si NCs possess surface facets while here almost spherical objects appear as Si cores in a SiO2 matrix. More interesting, however, is the significant reduction of the diameter variation of the gaps for Si NCs embedded in a SiO2 matrix. The corresponding fits give E g ≈ 0.44 eV + 1.69 eV/(d(nm))0.45 and E gex ≈ 0.44 eV + 1.78 eV/(d(nm))0.48 . Such a tendency for a weakening of the diameter dependence of the pair excitation energy has also been observed experimentally for three-dimensional arrangements of Si NCs embedded in amorphous SiO2 [8]. Fitting the experimentally measured confinement energies to the diameters yields exponents of 0.6 ± 0.1 (0.8 ± 0.1) for high-temperature (low-temperature) PL studies. On the other hand, it has also been shown that exponents around 1.3, as predicted within effective-mass models [39, 40], clearly seem to overestimate the diameter dependence. The pair excitation energies including many-body effects, E gex in figure 3 are only slightly larger than the Kohn–Sham HOMO–LUMO gaps E g . The differences are of the order or less than 0.1 eV. Moreover, the presence of oxygen seems to reduce additionally the difference ( E gex − E g ) compared to the values for hydrogenated NCs. The general reason for the small difference ( E gex − E g ) is the near compensation of quasiparticle and excitonic effects. A comparison with absolute values E gex for energies from other calculations is only possible for the hydrogenated Si NCs. In the range of NC diameter of 1– 2 nm the pair excitation energies from the SCF method are slightly smaller than the results from time-dependent density
functional theory (TDDFT) in the local-density approximation (TDLDA) [18] and also from quantum Monte Carlo (QMC) simulations by Puzder et al [17]. However, the diameter variations exhibit the same tendency [17, 19]. The calculated optical gaps in figure 3 can also be compared with measured excitation energies, e.g. from PL. However, such a comparison is difficult, because the origin of the measured emission lines is sometimes not very clear. Many data sets have been related to defect states or interface states localized in the interface region between the nanocrystal and matrix materials. In addition, there seems to be an influence of the Si NC synthesis in SiO2 , e.g. Si ion implantation into oxides [41], sputtering of Si rich oxides [4], or reactive evaporation of Si rich oxides [42]. Another problem concerns the definition and the determination of the NC diameter. For instance, Heitmann et al [8] have observed a strong blue shift of the PL signal from 1.3 to 1.65 eV with decreasing crystal size by using a mean crystal size between 1 and 6 nm. Their values are smaller than the calculated ones. Apart from the diameter definition, one mainly expects emission from the largest NCs with the smallest gap energies. In addition, the Stokes shifts present in the PL but not taken into account in figure 3 tend to reduce the pair excitation energies. The energy values in figure 3 also allow a separate discussion of the confinement energies [8] by subtracting the bulk gap value. In this way, the significant underestimation of the HOMO–LUMO gaps within DFT–LDA in the limit d → ∞ can be removed from the discussion. This also holds for the SCF method which gives the same gap value in the limit of vanishing confinement. For the NCs embedded in SiO2 the values resulting from figure 3 still tend to overestimate the experimental confinement energies [8]. The confinement energies, i.e., the difference between pair 4
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(a)
(b)
Figure 3. (a) HOMO–LUMO gaps E g and (b) pair excitation energies E gex versus NC diameter d . Red (gray) circles represent results for Si NCs embedded in amorphous SiO2 . For the comparison, energies for hydrogenated NCs are indicated as black circles. The dashed lines correspond to fits according to E g (E gex ) = E gbulk + α(nm/d)l .
Figure 4. Calculated probability distributions for (a) Si NCs embedded in a SiO2 matrix and (b) hydrogenated Si NCs for d = 1.0 nm. Red (gray) and blue (dark gray) isosurfaces represent HOMO and LUMO, respectively. Si core atoms in the NCs are indicated with green (light gray) balls.
energies and the bulk DFT–LDA gap, which amount to about 1.3–1.8 eV, are still larger than the experimental data [8]. While we calculate the gap openings due to confinement to be larger than 1 eV, the experimental values for dots with d ≈ 1 nm in a superlattice arrangement only amount to 0.6 eV [8]. However, in the case of other samples, mainly prepared by cosputtering of Si and SiO2 and subsequent annealing [43, 4–6], a tendency for similar or slightly higher confinement energies, as theoretically predicted, has been observed. Possible explanations have already been discussed for the absolute values of excitation energies above, e.g. the Stokes shift due to the atomic relaxation within the optically excited NCs. For hydrogenated Si NCs small Stokes shifts have been calculated [33] in agreement with the experimental observations [44]. Another explanation for these discrepancies could be a general underestimation of the sizes measured by xray diffraction (XRD) and using the Scherrer equation [8]. In addition, one cannot exclude that the experimental values are influenced by interface states and, hence, not only determined by the discussed quantum-confinement effects. A red shift due to electron–hole attraction could be another reason. An estimation gives values of the order of 0.1 eV. On the other hand, a comparison of calculated gaps and absorption onsets derived from measured spectra may be influenced by
the small magnitude of the oscillator strength for HOMO– LUMO transitions. A determination of the effective gap from absorption measurements may therefore give a larger value than the true lowest pair energy. The reason for the confinement reduction in the presence of the SiO2 matrix in comparison to the H-passivated case is verified by the representation of the square of the wavefunctions of Kohn–Sham orbitals at the top of the valence bands and at the bottom of the conduction bands for the model with d = 1.0 nm in figure 4(a). The wavefunction of the conduction state, the LUMO, is localized within the Si core of the NC, independent of the matrix. The electronic wavefunction of the HOMO extends more into the SiO2 matrix region. The changes in the probability densities for the hydrogenated NC as shown in figure 4(b) are obvious and drastic. In a hydrogenated NC the HOMO is mainly localized at the center of the NC while the LUMO is spread over a wider part of the core. Such a tendency has also been observed for free-standing Si NCs with an oxide overlayer [23]. The HOMO and LUMO wavefunctions for the Si NCs embedded in a SiO2 matrix are also completely different from those for NCs with 5
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Figure 5. Calculated site-projected electronic density of states (pDOS) onto Si atoms of Si nanocrystals embedded in SiO2 for dots with diameters of 1.0 and 1.4 nm. The pDOS for Si embedded in SiO2 is distinguished among the core (red/dark gray lines), suboxide interface (green/gray lines) and amorphous SiO2 (blue lines) regions. The pDOS onto Si atoms for hydrogenated Si nanocrystals is indicated by black lines. The Fermi level of energy zero of the computations is used as the middle of the energy gaps. Each spectrum is normalized so that the integrated DOS up to the Fermi level yields the same value.
Figure 6. Imaginary part of the frequency-dependent dielectric functions ε(ω) of (a)/(b) Si nanocrystals embedded in SiO2 and (c)/(d) hydrogenated Si nanocrystals with varying diameter: d = 0.8 nm (black lines), d = 1.0 nm (red/gray lines), d = 1.2 nm (green/light gray lines), d = 1.4 nm (blue/dark gray lines) and d = 1.6 nm (yellow/dotted lines). Different unit cell sizes are used for smaller and larger NCs. The arrows indicate the HOMO–LUMO gaps. The curves in (a), (c) have been computed for the 3 × 3 × 3 supercell while a 4 × 4 × 4 cell has been used for those in (b), (d).
partially oxidized Si NCs, more precisely saturation of two Si dangling bonds by an oxygen atom. Their HOMO and LUMO states are related to an O atom attached to a single silicon atom forming a double Si=O bond or an O atom connected to two silicon atoms creating a Si–O–Si bond at the surface [17, 19]. Finally, we analyze the electronic states away from the gap, especially the spatial character of the electronic states above the LUMO level and below the HOMO energy, using a site-projected density of states (pDOS) onto Si atoms. We distinguish three regions for Si atoms: core, interface, and SiO2 matrix. For the models with diameters of 1.0 and 1.4 nm, the pDOS illustrated in figure 5 shows a different behavior for the three regions. The density of states in figure 5 give three main pieces of information: (i) the band-gap regions are not only dominated by core but also by suboxide interface contributions independent of the NC size. Thereby, the interface contributions are stronger for the occupied part of the pDOS in agreement with figure 4(a). Matrix contributions are only observed far away (about 3 eV) from the HOMO and LUMO levels. (ii) Not only the HOMO–LUMO gap itself exhibits a strong variation with the NC size. This happens also for the entire energy interval with the strongly reduced density of states. The density gap decreases with increasing NC size as discussed above for the HOMO–LUMO gap as a consequence of quantum-confinement effects. (iii) In figure 5 we also compare the pDOS for Si NCs embedded in SiO2 with hydrogenated Si NCs. In all cases the gaps and regions of reduced DOS of H-passivated NCs are significantly larger compared to that of the embedded NCs as discussed above. This result is different from recent calculations performed for much smaller systems [45]. The line shapes of the DOS for the core region are similar for both dot types, for energies away from the fundamental gaps. The main effects of the missing host and the passivation of the surface dangling bonds
by hydrogen atoms is the apparent rigid shift of the occupied and empty states against each other. 3.2. Optical properties Results for the frequency-dependent dielectric function as obtained within the independent-particle approximation [34] are presented for both SiO2 -embedded (figures 6(a) and (b)) and hydrogenated (figures 6(c) and (d)) Si NCs of the same size and atomic geometry. More precisely, in figure 6 we present the imaginary parts of the frequency-dependent dielectric function for the three-dimensional, simple-cubic arrangement of Si NCs with varying diameter. Smaller (d = 0.8, 1.0 and 1.2 nm) and larger (d = 1.4 and 1.6 nm) NCs were constructed from 3 × 3 × 3 and 4 × 4 × 4 diamond Si unit cells, respectively. The intensity variation of spectra between figures 6(a) and (b), or figures 6(c) and (d), is mainly caused by the different filling factors f defined by the proportion of NC volume and total cell volume. The filling factors are f = 0.03, 0.06, 0.12, 0.08, and 0.12 for d = 0.8, 1.0, 1.2, 1.4 and 1.6 nm, respectively. Similar frequency dependences have been calculated for the dielectric function of both hydrogenated Si NCs [14, 15] and oxidized Si NCs [23–25]. The spectra shown in figure 6 represent those of individual NCs, at least in the energy range of confined states, i.e., below the onset of the absorption of the insulating matrix, and the independent (Kohn–Sham)-particle approximation. Local-field effects [18, 36] as well as excitonic couplings 6
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between adjacent NCs [8] have not been taken into account. The spectra of SiO2 -embedded and hydrogenated Si NCs are similar. They only show broad features with maxima around photon energies of 4.0–4.5 eV for the NCs. These maxima seem to develop into the E 2 peak of the bulk spectrum [14], whereas indications of the bulk E 1 peak are more or less not visible. One reason may be the neglect of excitonic effects which are important in bulk spectra [35]. However, even in the experimental spectra of Si NCs a corresponding peak is strongly suppressed. Confinement effects in this energy range are weak. There is a minor shift of the maxima towards smaller energies with increasing NC size. However, this trend is less pronounced in comparison to the strong confinement effects on the absorption edge. The spectra in figure 6, especially for the Si NCs embedded in the SiO2 matrix, exhibit optical absorption below the bulk E 0 transition of about 3.1 eV. This fact is clearly a consequence of breaking the bulk k-selection rule. The threshold of strong absorption does not coincide with the HOMO–LUMO gap [15, 46]. In other words, the indirect character of Si bulk is significantly reduced for the nanocrystalline structures, in particular after embedment in an amorphous matrix. In addition, the spectra of the embedded NCs are more smooth, while those of the hydrogenated ones exhibit several subpeaks and shoulders in agreement with the differences in the single-particle DOS in figure 5. The oscillator and spectral strengths of the spectra of the embedded Si NCs are surprisingly larger. One expects larger transition matrix elements for the NCs with stronger confinement, i.e., for the hydrogenated ones. The oscillator strength is inversely proportional to the characteristic volume. The reason for deviations from this rule is obvious from the DOS in figure 5. In the energy range of the broad spectral features there are contributions from the interface region which increase the imaginary parts of the dielectric functions. Therefore, we conclude that the interface region plays an active role in the optical properties of embedded NCs. Optical transitions in the ‘interface’ region do not occur in the spectral range of interest for hydrogenated NCs. The energies of the bonding and antibonding states belonging to Si–H bonds appearing far away from the energy region of interest. The general size dependence of the spectra roughly follows the number of electronic states and, hence, the number of Si atoms in the NCs. Therefore, the intensity of spectra grows with increasing NC size d . We must mention that the increase of Im ε(ω) with the NC diameter d seen in figure 6 is not a consequence of increasing oscillator strength, but rather of the increasing filling factor f . Recently, calculated absorption spectra have been published for other small Si clusters with 5 or 17 atoms in the core covered by one or two oxide shells or passivated by – OH group instead of hydrogen atoms [23]. These spectra show the reduced confinement effects in the presence of the oxide shells as similarly due to the presence of the SiO2 matrix. In addition, the discussed enhancement of the peak intensities is clearly visible. In [23] it has been demonstrated that this effect is a consequence of spatial contributions from electronic states outside the Si core which contribute to the total oscillator strength.
Several experimental studies of optical properties of Si NCs embedded in a SiO2 matrix have been reported. Gallas et al [9] have determined optical properties in the 1.6–6.2 eV spectral range using nanocrystals with diameters of 1 and 4.5 nm. Ding et al [10, 11] also determined optical properties in the energy range of 1.1–5.0 eV using nanocrystals with an average size of ∼4.2 nm and reported that the dielectric functions of Si NCs embedded in SiO2 show a significant suppression as compared to bulk Si. In all these experiments [9–11], spectroscopic ellipsometry has been applied. In combination with model assumptions, either the validity of the Bruggeman effective medium approximation [47] or the Forouhi–Bloomer model [48], one was able to extract the imaginary part of the NC dielectric function in a wide spectral range from the raw data. Contradictory experimental results have been published. For diameters of 4.6– 7.6 nm almost bulk-like spectra with E 1 and E 2 peaks have been observed by Ding et al [10, 11]. However, the strength of the main E 1 and E 2 peaks exhibits a large reduction and a significant redshift of the E 2 structure depending on the NC size. In the light of our results these findings are somewhat surprising. We have no clear explanation. In figure 6 we find indications for such a redshift only for the smallest cluster studied. On the contrary, Gallas et al [9] found that the dielectric functions of the large Si NCs with diameters close to 4.5 nm show spectral features localized at photon energies h¯ ω = 3.5, 4.5 and 5.2 eV ascribed to the bulk E 1 , E 2 and E 1 critical points, respectively. For the smaller Si clusters with diameter close to 1 nm, the structure associated with the E 1 critical point, near 3.5 eV, disappeared while those at higher energies, near E 2 and E 1 , remained present. These results are widely in agreement with our calculations in figures 6(a) and (b), which show a pronounced E 2 -like feature and an indication for a shoulder at higher energies which may be interpreted as the E 1 -like structure in the bulk case. The observed differences between figure 6 and experimental spectra (especially of those of Ding et al [10, 11]) may be traced back to effects due to the arrangement of the Si NCs in the SiO2 matrices, i.e., certain local-field effects.
4. Conclusions In conclusion, we have studied the influence of embedding amorphous SiO2 matrices on the properties of Si nanocrystals using state-of-the-art total energy and electronic-structure calculations. In order to determine the lowest pair excitation energies of the nanocrystal systems, which include the Coulombic electron–hole interactions and quasiparticle effects, we have also carried out SCF calculations. Our calculations show the importance of quantum-confinement effects, not only for hydrogenated nanocrystals, but also for Si nanocrystals embedded in an amorphous SiO2 matrix. We found significant differences in the electronic and optical properties between hydrogenated and oxidized nanocrystals. The presence of an amorphous matrix leads to drastic reductions of the absolute pair excitation energies and also of their dependence on the NC diameter. In agreement with recent measurements we have predicted a size dependence 1/d n of the confinement 7
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contribution to the pair energy with n < 1. The amorphous SiO2 matrix yields a drastic reduction of the electronic confinement effects near the band-gap and, hence, the onset of the optical absorption. We have traced these findings back to the HOMO (and also LUMO) wavefunctions also extending into the interface and matrix regions. The resulting confinement energies seem to overestimate the values derived from a part of the photoluminescence measurements but are smaller than predictions extracted from absorption onsets. A possible explanation has been discussed in the text. Studying optical absorption spectra, we have clearly demonstrated the break of the indirect Si character below photon energies of the order of the bulk E 0 transition, especially in the presence of an amorphous matrix. The confinement effects in the higher energy regions of optical transitions are reduced. The computed spectra show a pronounced maximum near the bulk E 2 transition energy. These findings are in reasonable agreement with the broad peak of the frequencydependent dielectric function recently measured by means of spectroscopic ellipsometry. Differences between calculated and measured spectra have been interpreted in terms of possible strong local-field effects in the optical spectra due to the three-dimensional arrangement of the nanocrystals.
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Acknowledgments This work was supported by the BMBF, Germany (Project Nos. 03SF0308 and 13N9669) and by the DFG (within SPP1181, project KR1805/8-1 and through a Heisenbergfellowship, KR1805/9). Grants of computer time from John von Neumann Institute for Computing (NIC) in J¨ulich and from the Texas Advanced Computing Center (TACC) in Austin, TX, are gratefully acknowledged.
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