Optical Emission of Statistical Distributions of Silicon Quantum Dots

Article pubs.acs.org/JPCC

Optical Emission of Statistical Distributions of Silicon Quantum Dots A. S. Barnard*,† and H. F. Wilson†,‡ †

CSIRO Virtual Nanoscience Laboratory, 343 Royal Parade, Parkville 3052, Victoria, Australia School of Applied Sciences, RMIT University, Swanston Street, Melbourne 3000, Victoria, Australia

‡

ABSTRACT: Silicon quantum dots are a versatile luminescent material that can be easily integrated with existing Si-based optoelectronic devices, and with improved control over the optical emission and spectral resolution, they will facilitate new applications in other domains. Although the creation of size-and shape-selected silicon quantum dots with tunable properties is highly desirable, reducing the size (to access lower wavelengths) typically comes at a cost of increasing the bandwidth. However, using a polydispersed ensemble of silicon quantum dots (and some simple statistical models), we find that although spectral resolution can be improved by targeting specific shapes, achieving perfect monodispersivity is unnecessary. Depending on the optical properties required, simply restricting the polydispersivity of the sample in the right way may be sufficient, irrespective of the statistical distribution present in the sample.

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INTRODUCTION

In this study, we combine a large number of explicit electronic structure simulations and some simple statistical models to explore the size- and shape-dependence of the peak wavelength for the characteristic optical emission of ensembles of hydrogen-passivated SiQDs. We find that there are likely to be definite advantages to attempts to produce or separate SiQDs with particular sizes and shapes, to shift the peak wavelength or to reduce the bandwidth. By predicting quality factors for a variety of cases, we find that there is a clear motivation for targeting cubic SiQDs to increase the selectivity and efficiency of the luminescence. A slightly rounded cube, with the edges and corners truncated, is predicted to have the highest spectral resolution, regardless of the statistical distribution in particle size, and perfect monodispersivity is not essential to see rewards.

Silicon quantum dots (SiQDs) are an extremely attractive material for a broad range of optoelectronic applications. Silicon’s high abundance on Earth and apparent low toxicity in nanoparticle form1 as well as the enormous existing industrial expertise and investment in silicon-based technology make it an ideal alternative to other quantum dot materials such as CdSe and PbS. SiQDs may be readily fabricated by a wide variety of means in solid-, liquid-, gas-, and plasma-phase methods,2 exhibiting a variety of size and shape distributions3−5 depending on formation method and conditions.6,7 The adoption of SiQDs in many optoelectronic applications, however, lags behind other less desirable materials such as CdSe and PbS because of the difficulty of obtaining emission across the entire visual spectrum with a narrow bandwidth. SiQD-based LEDs were first reported8 in the near-infrared, and then extended into the visible red range9,10 and recently the yellow-green region.11 Unfortunately, obtaining SiQDs whose emission extends into the green and blue regime has so far remained elusive. Enhanced control over the optical properties of SiQDs and their distributions should allow new technological applications for silicon as an optoelectronic material. Although it is possible to simulate functional properties such as the optical emission spectrum of a given material, previous works on chemical systems have shown that the values are unique to each structure and sensitive to isomeric variations.12 This does not pose a problem in samples that can be purified, but (like many nanoparticle systems) SiQDs samples cannot. Structural polydispersivity is always present in as-grown samples, so it is far more useful to simulate the ensemble average of these properties, using a virtual sample that mimics diversity of real specimens. Fortunately, this too can be achieved using efficient theoretical and computational methods, combined with appropriate statistical modeling.13,14 Published 2015 by the American Chemical Society

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METHODS The data set used in this study contains 302 hydrogenterminated SiQDs with a diameter between 0.5 and 3 nm and a large range of different morphologies defined by zonohedrons enclosed by {100}, {110}, {111}, and {113} facets. These are the four lowest-energy H-terminated surface facets and include the range of lowest-energy morphologies.6 The data set is complete in the sense that it represents all possible silicon structures which can be formed by symmetric cleaving along any combination of these planes around the centrosymmetric lattice site. SiQDs in the data set are labeled by their shapes on the basis of the fraction of each surface facet which they present. Pure {100}-, {110}-, {111}-, and {113}-faceted SiQDs are labeled cubes (C), octahedra (OH), rhombic dodecahedra (RD), and deltoidal icositetrahedra (DI), respectively. SiQDs Received: February 5, 2015 Revised: March 15, 2015 Published: March 26, 2015 7969

DOI: 10.1021/acs.jpcc.5b01235 J. Phys. Chem. C 2015, 119, 7969−7977

The Journal of Physical Chemistry C

Article

bulk silicon and hydrogen in a gas of H2. Within a given sizeselected ensemble, larger structures and lower energy shapes have a higher probability of observation and therefore contribute more to the properties of the entire sample. Because larger nanoparticles have smaller formation energy up to the bulk limit, the Boltzmann distribution is only well-defined for a size-selected nanoparticle distribution. Alternatively, if the synthesis and/or stability is not thermodynamically limited, then a range of other distributions are possible, or could be deliberately encouraged. This includes a normal distribution over the size range

presenting two facets are named with the base name of the majority facet and described as being truncated by the other. SiQDs with three different facets are labeled as doubly truncated versions of the shape described by the majority facet. No SiQD in the data set presents more than three surface facets. All SiQD geometries in the data set were terminated with hydrogen so that each silicon atom was tetrahedrally 4fold-coordinated prior to relaxation. The (unrelaxed) geometries of all SiQDs in the data set are freely available on the CSIRO Data Access Portal.15 Because this study involves so many simulations, we have used the density functional tight-binding method with selfconsistent charges (SCC-DFTB),16,17 which was implemented in the DFTB+ code,18 to perform the individual calculations. This is an approximate quantum chemical approach where the Kohn−Sham density functional is expanded to the second order around a reference electron density. The reference density is obtained from self-consistent density functional calculations of weakly confined neutral atoms within the generalized gradient approximation (GGA). The confinement potential is optimized to anticipate the charge density and effective potential in molecules and solids. A minimal valence basis set is used to account explicitly for the two-center tightbinding matrix elements within the DFT level. The double counting terms in the Coulomb and exchange−correlation potential as well as the intranuclear repulsion are replaced by a universal short-range repulsive potential. All structures have been fully relaxed with a conjugate gradient methodology until forces on each atom was minimized to be less than 10−4 a.u. (i.e., ∼5 meV/Å). In all the calculations, the PBC set of parameters is used to describe the contributions from diatomic interactions of silicon.19 This method has been shown to provide good and reliable results for SiQDs in the past.20 Because we are concerned with the properties of a diverse ensemble of possible structures, we have analyzed expectation values. The expectation value of observable is defined as

pi =

1 Z

1 2 2πσsize

⎛ (D − ⟨λ⟩ )2 ⎞ size ⎟ exp⎜ − i 2 2σsize ⎠ ⎝

(4)

or a Poisson distribution over the size range pi =

Di 1 ⟨λ⟩size exp( −⟨λ⟩size ) Z Di!

(5)

where Z is the partition function over all of the structures i, ⟨λ⟩size is the average size of the SiQDs in the ensemble, Di is the spherically averaged diameter of each i, and σsize is the variance of the size distribution of the ensemble. In addition to this, the SiQDs may have a random distribution, which many be simulated with a random number generator, or a frequency distribution, where the probability of each structure is treated independently and is determined a priori upon construction of the ensemble. This provides us with five different types of distributions that may be relevant to different synthesis and processing conditions, but in principle, any other distribution could be applied. A graph showing how these distributions relate to the present data set (as a function of the particle size) is provided in Figure 1.

n

⟨λ⟩ =

∑ px i i i=1

(1)

and the associated variance, σ2, is defined as n

σ2 =

∑ pi (xi − ⟨λ⟩)2 i=1

(2)

Both parameters are calculated by summing over the individual properties x of all structures i. The total number of structures for each set is n, and pi is the probability of observation of i. There are numerous ways to define the probability on the basis of different statistical distributions. One possibility is a thermodynamic distribution pi =

exp−ΔGi /(kBT ) n

∑i = 1 exp−ΔGi /(kBT )

(3)

where kB is Boltzmann’s constant and the denominator is the canonical partition function. The change in the free energy ΔGi = ∑j(NjEj − Ei(j)) describes the thermodynamic stability (with respect to the reservoir and bulk silicon reference state), as a function of the total energy, Ei(j), of particle i, containing j chemical species; Nj represents the number of atoms of species j, and Ej is the energy of j in the reservoir. This can be defined with respect to any chemical reservoir (temperature and/or supersaturation) that is required, but in this case, we have used

Figure 1. Comparison of the probability distributions compared in the study, as a function of the diameter of the SiQDs in the data set. 7970



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Figure 2. (a) Computed formation energy per silicon atom of the full silicon nanoparticle data set as a function of effective radius. (b) Computed DFTB band gap as a function of effective radius (defined as the radius of a sphere of bulk silicon density with the same number of silicon atoms) for the full hydrogen-terminated silicon nanoparticle data set. (c) The computed DFTB band gap as a function of effective radius, corrected with respect to the baseline polynomial to illustrate the degree of spectral scatter.

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of facets (and hence the degree of “sphericity”) for a given volume. Although the trends in the raw data are clearly evident, the polydispersivity in the size and shape of the sample gives rise to considerable scatter in the results, which will increase the bandwidth of mixed sample. The scatter is depicted more clearly by Figure 2c, which depicts the relative wavelength of each SiQD compared to an average polynomial fit. It may be seen that cubic SiQD uniformly exhibit substantially larger wavelengths, and octahedral nanoparticles substantially smaller wavelengths, than those of the average nanoparticle of a given size, consistent with earlier results.20 Truncated SiQDs exhibit wavelengths lying between the cubic and octahedral extremes. This spread becomes evident when we calculate the ensemble average (the central wavelength) and the associated

DISCUSSION OF RESULTS

Formation energies and computed DFTB band gaps as a function of SiQD shape and size for the full data set are plotted in Figure 2a,b. All formation energies are given relative to bulk silicon and a molecular hydrogen reservoir with a chemical potential μ = 0; changing the value of μ changes the energetics of different SiQD shapes.6 Results of the full data set have been made openly available online.21 We find a clear dependence of the band gap upon both SiQD size and shape. Smaller SiQDs are known to exhibit larger band gaps because of quantum confinement effects;22 however, the additional relationship between SiQD shape and size cannot be explained solely by quantum confinement because quantum confinement would lead us to expect a wavelength that increases with the number 7971



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Figure 3. (a) Predicted optical emission spectrum of the entire unrestricted ensemble of SiQDs. (b) Shift in the emission wavelength as a function of SiQD size. (c) Change in the spectral resolution (Q factor) as a function of SiQD size. In each case, results for different statistical distributions are provided.

distributions. This suggests that thermodynamically controlled synthesis may be sufficient to tune the spectrum, even if no attempt to control the size and/or shape of SiQDs was made. In general, the predicted spectra are quite broad, which is, of course, related to the scatter displayed in Figure 2c. To identify which SiQDs are most responsible for the bandwidth, the emission for individual SiQDs has been corrected with respect to the ensemble average in Figure 3b for each of the different distributions described above. Here we can see that because there is a greater number of larger SiQDs in the ensemble, it is the small SiQDs that increase the spectral bandwidth. This relationship is largely independent of the statistical distribution but is slightly more significant for the room-temperature Boltzmann distribution. Creation of size-selected nanoparticle ensembles is possible by various means. As-formed nanoparticle distributions often

variance and use a Gaussian function to estimate the emission spectrum. Because the spectrum emanating from each particle is unique and additive, the spectrum emanating from the entire sample depends on the structural diversity of the sample, i.e., the statistical distribution. In Figure 3a, the spectrum is shown for the Boltzmann distribution (at T = 1373 K and 300 K), the size-dependent normal distribution, the size-dependent Poisson distribution, the frequency distribution, and a random distribution (on the basis of a pseudorandom number generator). The numerical values are provided in Table 1. Here we can see how the spectrum depends on the SiQD distribution, both in terms of spectral position and resolution. The room-temperature Boltzmann and the size-dependent normal distributions exhibit greater intensity, though the Boltzmann is red-shifted. At high temperature, the Boltzmann distribution is similar to the Poisson, frequency, and random 7972



7973

ΔQ Δ⟨λ⟩

−32.71 −22.81 −29.58 −23.84 +5.84 −1.23 +2.85 +15.48 +8.27 +9.87 +20.90 +31.97 −19.59 −31.29 +9.58 −14.96 −7.16 +7.51 +42.20 −8.68 +30.65 +12.26 +13.87 +9.29

ΔQ Δ⟨λ⟩

−21.97 −18.65 −20.24 −23.35 +7.58 −8.05 −0.27 +8.70 +9.91 +4.26 +5.57 +11.69 −25.91 −33.18 +16.62 −15.47 −1.95 +39.87 +80.46 −3.96 +32.35 +36.03 +26.01 +45.70

a

ΔQ Δ⟨λ⟩

−25.80 −16.87 −30.79 −19.27 +9.10 −2.72 −0.63 +16.71 +6.01 +7.58 +15.75 +28.59

ΔQ

The results for different statistical distributions are provided, where the +(−) sign indicates a blue(red)shift in the spectrum, and an increase (decrease) in spectral resolution (i.e., sharper bandwidth). In this case, no attempt has been made to control the size. Note that some shapes (the doubly truncated rhombic dodecahedron, deltoidal icosahedrally truncated cube, and the doubly truncated octahedron) appear in the unrestricted ensemble but in insufficient numbers to allow for a separate shape-dependent ensemble.

−49.85 −37.62 −31.21 −15.31 +0.89 −5.47 +9.20 +21.78 +12.21 +5.23 +24.82 +32.75

Δ⟨λ⟩ ΔQ

−31.36 −29.71 +27.86 −11.90 −9.79 +62.88 +111.09 −12.22 +36.66 +56.30 +34.54 +68.54 −41.25 −31.82 −27.28 −22.73 +5.03 +5.50 +10.06 +10.91 +11.66 +16.25 +22.25 +36.66

Δ⟨λ⟩ ΔQ

−27.87 −31.00 +25.17 −13.86 −6.05 +47.58 +88.97 −11.28 +30.78 +42.40 +26.79 +48.13

random

−10.11 −23.07 +25.07 −14.95 +4.46 +19.16 +45.35 +12.84 +47.26 +13.41 +18.65 +25.34

have a broad range of sizes, but several schemes exist for separating out nanoparticles of a given size. For silicon nanoparticles, density-gradient ultracentrifugation24 has been demonstrated, with nanoparticles suspended in a fluid arranging themselves according to density in a high-speed centrifuge. Size-selective precipitation has also been demonstrated,11,23 with nanoparticles suspended in toluene and precipitated out using methanol as an antisolvent; the largest nanoparticles precipitate out first. This method was used to narrow size distributions for the creation of multicolored SiQD LEDs.11 Another way of assessing the spectral resolution is to predict the quality factor (Q-factor) of the emission, in addition to the red- or blue-shifts associated with any tuning attempts. This is a dimensionless parameter used throughout physics and engineering to describe efficiency, particulary for devices such as oscillators and resonators,25 where it is defined as the amount of stored mechanical energy divided by some measure of the friction, damping, resistance, or other form of energy dissipation (energy loss). It is typically used to characterize the qualitative behavior of wide range of optical, electrical, and electromechanical systems. For example, in signal processing, the Q-factor is determined by the resonant frequency divided by the bandwidth (full width half-maximum, fwhm), f 0/Δf. By analogy, our Q-factors can be obtained dividing the expectation value by the standard deviation, ⟨λ⟩/(σ2)1/2. The predicted Q-factors for individual SiQDs and the entire unrestricted ensemble are provided in Figure 3c and Table 1, respectively, for each of the distributions. Here we can see that in general larger SiQDs show an increase in the Q-factor (spectral resolution) and smaller SiQDs show a decrease in spectral resolution. Interestingly, however, this does depend on the distribution: At larger sizes, a size-dependent normal distribution provides a greater improvement than the alternatives, but at small sizes, a frequency or random distribution are likely to limit the spectral degradation. In general, it is possible to tune the wavelength of SiQD emission, but this comes at the cost of increasing the bandwidth at the blue end of the spectrum. In each of the cases discussed above, no attempt has been made to modify the shape of the SiQDs, but this is possible experimentally. Silicon quantum dots may be produced via different methods with various shapes including cubes,5 octahedra,3 truncated octahedra,4 and, most commonly, pseudospheres, depending on production method and thermodynamic conditions. Recently, we have computed optimal morphologies for freestanding hydrogen-terminated

−7.23 +14.79 −32.50 −8.64 +10.59 −11.38 −16.76 +19.02 −4.03 −4.00 +6.99 +20.05

The results for different statistical distributions are provided.

Δ⟨λ⟩

13.80 9.71 12.75 8.72 8.02 7.90

shape

44.87 61.38 46.53 67.16 72.69 73.83

rhombic dodecahedron (RD) deltoidal icositetrahedron (DI) DI-truncated cube octahedron (OH) cube (C) OH-truncated RD doubly truncated cube DI-truncated OH OH-truncated cube RD-truncated cube RD-truncated octahedron DI-truncated RD

619.17 596.18 593.67 585.47 583.16 583.10

frequency

Boltzmann (300 K) Boltzmann (1373 K) normal Poisson frequency random

Poisson

Q-factor

normal

(σ2)1/2 (nm)

Boltzmann (1373 K)

⟨λ⟩ (nm)

Boltzmann (300 K)

a

distribution

Table 2. Shift in the Peak Position (Δ⟨λ⟩, in nm) and Spectral Resolution (ΔQ, in %) with Respect to the Entire Unrestricted Ensemble Containing 302 Hydrogen-Terminated SiQDs with a Diameter between 0.5 and 3 nm and a Large Range of Different Morphologies Defined by Zonohedrons Enclosed by {100}, {110}, {111}, and {113} Facetsa

Table 1. Ensemble Average (⟨λ⟩), Standard Deviation ((σ2)1/2), and Spectral Resolution (Q-factors) for the Entire Unrestricted Ensemble Containing 302 HydrogenTerminated SiQDs with a Diameter between 0.5 and 3 nm and a Large Range of Different Morphologies Defined by Zonohedrons Enclosed by {100}, {110}, {111}, and {113} Facetsa

−34.08 −30.93 +14.59 −8.75 −20.60 +60.36 +117.92 +3.46 +32.14 +44.77 +52.46 +66.23

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Figure 4. Predicted optical emission spectrum for samples restricted (grown or filtered) to enrich the presence of the (a) {100}, (b) {110}, (c) {111}, and (d) {113} facets, comparing the impact of different statistical distributions.

SiQDs via density functional theory as a function of thermodynamic conditions6,7 and shown that shapes including subsets of the (111), (100), and (113) facets may be formed depending on thermodynamic conditions, consistent with observations of shapes obtained via gas-phase pyrolysis of silane,3,4 whereas plasma-based methods5 tend to form cubic shapes that maximize the amount of surface hydrogen absorption. However, most methods for fabrication have been found to produce pseudospherical SiQDs that may not represent a thermodynamic ground state but rather are merely the end point of a kinetically limited growth process. Annealing of these pseudospherical SiQDs in an appropriate atmosphere may produce faceted SiQDs of appropriate shapes, although this has yet to be demonstrated in practice.

Because shape control requires significant intervention, and monodispersivity still remains elusive, this invites the question: Is it likely to be worth the effort? Provided in Table 2 is the shift in the peak position (Δ⟨λ⟩) and spectral resolution (ΔQ) that may be expected if different shape-selected samples were obtained (either grown or filtered postsynthesis), with respect to the entire unrestricted ensembles with different distributions. All results are given with respect to the entire sample because the absolute values are highly dependent on the contents of each subset, but the values in this table should still be used with caution. The usefulness of this information lies in comparing the impact of different distributions as well as different shapes. For example, we can see that regardless of the type of statistical distribution in size certain shapes consistently provide an 7974



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Figure 5. Predicted optical emission spectrum for samples restricted (grown or filtered) to suppress the presence of the (a) {100}, (b) {110}, (c) {111}, and (d) {113} facets, comparing the impact of different statistical distributions.

increase in the spectral resolution (ΔQ = +%) whereas others consistently show a reduction in spectral resolution (ΔQ = −%). In particular, the doubly truncated cube (which appears as a slightly “rounded” cube, with the corners truncated perpendicular to ⟨111⟩ and the edges truncated perpendicular to ⟨110⟩) shows an increase in spectral quality between 42% and 118%, with almost no change in the emission wavelength unless the SiQDs have a Boltzmann distribution at low energy. At this point, it is wise to point out that there are a limited number of these structures in the ensemble, so a larger study would be useful to confidently eliminate the possibility of artifacts associated with the size of the sample. However, as stated above, this degree of shape selectively is still challenging, so it is prudent to consider alternative

situations where some mixing of different morphologies is still permitted. Taking a more coarse-grained approach, it is possible to create a range of restricted ensembles where the SiQDs have been enriched with facets in different orientations but the shape-selectivity is still imperfect. With this aim in mind, we have recalculated the ensemble average and variance for samples that have been enriched with {100}, {110}, {111}, and {113} facets. In this context, enriched particles are those with >50% of the available surface area in a given {hkl} orientation. Figure 4a−d reproduces the emission spectrum presented for the unrestricted ensemble in Figure 3a, for the {100}-, {110}-, {111}-, and {113}-enriched samples, respectively. Here we can see that there is a distinct advantage in enriching a sample with {100} facets and a distinct disadvantage 7975



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(3) Murthy, T. U. M. S.; Miyamoto, N.; Shimbo, M.; Nishizawa, J. Gas-Phase Nucleation During the Thermal Decomposition of Silane in Hydrogen. J. Cryst. Growth 1976, 33, 1−7. (4) Körmer, R.; Butz, B.; Spiecker, E.; Peukert, W. Crystal Shape Engineering of Silicon Nanoparticles in a Thermal Aerosol Reactor. Cryst. Growth. Des. 2012, 12, 1330−1336. (5) Mangolini, L.; Kortshagen, U.; Bapat, A. Plasma Synthesis of Semiconductor Nanocrystals for Nanoelectronics and Luminescence Applications. J. Nanopart. Res. 2006, 2007, 39−52. (6) Wilson, H. F.; Barnard, A. S. Predictive Morphology Control of Hydrogen-Terminated Silicon Nanoparticles. J. Phys. Chem. C 2014, 118, 2580−2586. (7) Wilson, H. F.; Barnard, A. S. Thermodynamic Control of Halogen-Terminated Silicon Nanoparticle Morphology. Cryst. Growth. Des. 2014, 14, 4468−4474. (8) Cheng, K.-Y.; Anthony, R.; Kortshagen, U. R.; Holmes, R. J. Hybrid Silicon Nanocrystal- Organic Light-Emitting Devices for Infrared Electroluminescence. Nano Lett. 2010, 10, 1154−1157. (9) Puzzo, D. P.; Henderson, E. J.; Helander, M. G.; Wang, Z.; Ozin, G. A.; Lu, Z. Visible Colloidal Nanocrystal Silicon Light-Emitting Diode. Nano Lett. 2011, 11, 1585−1590. (10) Cheng, K.-Y.; Anthony, R.; Kortshagen, U. R.; Holmes, R. J. High-efficiency Silicon Nanocrystal Light-Emitting Devices. Nano Lett. 2011, 11, 1952−1956. (11) Maier-Flaig, F.; Rinck, J.; Stephan, M.; Bocksrocker, T.; Bruns, M.; Kübel, C.; Powell, A. K.; Ozin, G. A.; Lemmer, U. Multicolor Silicon Light-Emitting Diodes (SiLEDs). Nano Lett. 2013, 13, 475− 480. (12) Sattler, K. The energy gap of clusters, nanoparticles, and quantum dots. In Handbook of Thin Film Materials; Nalwa, H. S., Ed.; Academic: San Diego, CA, 2002; Vol. 5, pp 61−97. (13) Lai, L.; Barnard, A. S. Tuning the Electron Transfer Properties of Entire Nanodiamond Ensembles. J. Phys. Chem. C 2014, 118, 30209−30215. (14) Shi, H. Q.; Rees, R. J.; Per, M. C.; Barnard, A. S. Impact of Distributions and Mixtures on the Charge Transfer Properties of Graphene Nanoflakes. Nanoscale 2015, 7, 1864−1871. (15) Barnard, A.; Wilson, H. Silicon Nanoparticle Structure Set, v1; CSRIO Data Collection. CSIRO Australia: Clayton South, Victoria, Australia, 2014. DOI: 10.4225/08/546AA009190C4. (16) Porezag, D.; Frauenheim, Th.; Köhler, Th.; Seifert, G.; Kaschner, R. Construction of Tight-Binding-Like Potentials on the Basis of Density-Functional Theory: Application to Carbon. Phys. Rev. B 1995, 51, 12947−12957. (17) Frauenheim, Th.; Seifert, G.; Elstner, M.; Niehaus, Th.; Köhler, C.; Amkreutz, M.; Sternberg, M.; Hajnal, Z.; di Carlo, A.; Suhai, S. Atomistic Simulations of Complex Materials: Ground-State and Excited-State Properties. J. Phys.: Condens. Matter 2002, 14, 3015. (18) Aradi, B.; Hourahine, B.; Frauenheim, Th. DFTB+, a Sparse Matrix-Based Implementation of the DFTB Method. J. Phys. Chem. A 2007, 111, 5678−5684. (19) Köhler, C.; Frauenheim, T. Molecular Dynamics Simulations of CFx (x = 2. 3) Molecules at Si3N4 and SiO2 Surfaces. Surf. Sci. 2006, 600, 453−460. (20) Wilson, H. F.; McKenzie-Sell, L.; Barnard, A. S. Shape Dependence of the Band Gaps in Luminescent Silicon Quantum Dots. J. Mater. Chem. C 2014, 2, 9451−9456. (21) Barnard, A.; Wilson, H. Silicon Quantum Dot Data Set, v1; CSIRO Data Collection; CSIRO Australia: Clayton South, Victoria, Australia, 2015. DOI: 10.4225/08/55060C818BF0C. (22) Ledoux, G.; Gong, J.; Huisken, F.; Guillois, O.; Reynaud, C. Photoluminescence of Size-Separated Silicon Nanocrystals: Confirmation of Quantum Confinement. Appl. Phys. Lett. 2002, 80, 4834−4836. (23) Mastronardi, M. L.; Maier-Flaig, F.; Faulkner, D.; Henderson, E.; Kübel, C.; Lemmer, U.; Ozin, G. A. Size-Dependent Absolute Quantum Yields for Size-Separated Colloidally-Stable Silicon Nanocrystals. Nano Lett. 2012, 12, 337−342. (24) Mastronardi, M. L.; Hennrich, F.; Henderson, E. J.; Maier-Flaig, F.; Blum, C.; Reichenbach, J.; Lemmer, U.; Kübel, C.; Wang, D.;

in enriching a sample with {113} facets regardless of the statistical distribution. This is consistent with expectation, on the basis of the contents of Table 2. Inverting the problem, we can also investigate the impact of (imperfect) attempts to suppress certain crystallographic facets. In this context, suppressed particles are those with

Optical Emission of Statistical Distributions of Silicon Quantum Dots

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