Eimear
Madden
and
Martijn A.
Zwijnenburg
*
Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, UK. E-mail: m.zwijnenburg@ucl.ac.uk
First published on 21st March 2024
We use a combination of many-body perturbation theory and time-dependent density functional theory to study the optical and electronic properties of hydrogen terminated silicon nanoparticles. We predict that the lowest excited states of these silicon nanoparticles are excitonic in character and that the corresponding excitons are completely delocalised over the volume of the particle. The size of the excitons is predicted to increase proportionally with the particle size. Conversely, we predict that the fundamental gap, the optical gap, and the exciton binding energy increase with decreasing particle size. The exciton binding energy is predicted to counter-act the variation in the fundamental gap and hence to reduce the variation of the optical gap with particle size. The variation in the exciton binding energy itself is probably caused by a reduction in the dielectric screening with decreasing particle size. The intensity of the excited state corresponding to the optical gap and other low energy excitations are predicted to increase with decreasing particle size. We explain this increase in terms of the ‘band structure’ becoming smeared out in reciprocal space with decreasing particle size, increasing the ‘overlap’ between the occupied and unoccupied quasiparticle states and thus, the oscillator strength. Fourier transforms of the lowest excitons show that they inherit the periodicity of the frontier quasiparticle states. This, combined with the delocalisation of the exciton and the large exciton binding energy, means that the excitons in silicon nanoparticles combine aspects of Wannier–Mott, delocalisation and effect of periodicity of the underlying structure, and Frenkel, large exciton binding energy, excitons.
Because of the covalent character of the bonding in silicon, undercoordinated silicon atoms on the surface of the SiNPs need to be terminated by capping groups. These capping groups can, for example, be hydrogen atoms, hydroxyls or organic groups. Experimentally, the exact nature of the capping group depends on the synthesis conditions. In the case of organic groups, they are generally specifically introduced during synthesis, for example, so that the synthesised SiNPs have a special affinity to a biomolecule the SiNP will be used to sense.2 Computationally, mostly hydrogen terminated SiNPs, the most general of SiNPs, are studied under the implicit assumption that the perturbation induced by capping agents other than hydrogen on the optical and electronic properties of the silicon core of the SiNPs are small and that hydrogen terminated SiNPs are thus good models of SiNPs terminated with other capping agents. Additionally, SiNPs can take different shapes, corresponding to different cuts from the bulk silicon structure,4 the most common studied of which are spherical and octahedral SiNPs.
The key optical and electronic properties of SiNPs include the fundamental gap, the optical gap and the exciton binding energy (see Fig. 1). The fundamental gap is the energy required to excite a non-interacting electron–hole pair. The optical gap is the energy required to instead excite an interacting electron–hole pair, often referred to as an exciton, which is bound by the mutual Coulombic interaction between electron and hole. For particles with a singlet electronic ground state like the SiNPs this optical gap corresponds to the lowest energy singlet excitation. The exciton binding energy, finally, is a measure of how much the exciton is stabilised relative to a non-interacting electron–hole pair, and the difference between the fundamental and optical gap (see Fig. 1). Analytical models for the electronic and optical properties of semiconductor nanoparticles based on the effective mass approximation (EMA),5 or using particle in a sphere wavefunctions as basis,6 predict that the fundamental gap of such particles varies as 1/r2 with the particle size r and that the exciton binding energy, or at least its Coulombic part, varies as 1/r. The optical gap, as the fundamental gap minus the exciton binding energy, is predicted by the same models to vary as 1/rn, where n varies with particle size but always lies somewhere between 2 and 1. Analytical models, however, always include necessary approximations. Numerical quantum chemical calculations that make less assumptions yield n values that deviate from the values predicted by the analytical models.7–10 For example, in the case of the fundamental gap such calculations predict n values ranging from 0.7 to 1.5 instead of 2.
Changes in the SiNP size impact not only the fundamental and optical gap, and the exciton binding energy. Various other electronic and optical properties of the SiNPs also change with particle size. For example, with the optical gap, the whole UV-Vis absorption spectrum of SiNPs shifts to the blue with decreasing particle size. The photoluminescence maxima of SiNPs have also been predicted to shift with decreasing particle size to larger energies/shorter wavelengths.11–15 Similarly, the associated Stokes shift, the difference in energy between the optical gap and the photoluminescence maxima, is predicted to increases with decreasing particle size.11–15 Finally, the oscillator strength of the excitations near the optical gap was predicted to increase with decreasing particle size, which was linked in the literature to a transition from an indirect to a direct ‘band gap’ upon nanostructuring of the silicon.16,17
Because of the fundamental interest in SiNPs as model systems for semiconductor quantum dots, as well as the technological applications of SiNPs, hydrogenated silicon nanoparticles have been studied using a wide range of theoretical approaches, including tight-binding models,8,16,18–20 empirical pseudopotential theory,9,21–23 (tight-binding) density functional theory (DFT),7,13,17,24–27 (tight-binding) time-dependent DFT (TDDFT),11,12,14,15,28–36 quantum Monte Carlo methods,35,37 and many-body perturbation theory in the form of GW and GW in combination with solving the Bethe Salpter Equation (GW-BSE).10,31,38–41 However, few of these studies consider the combination of the fundamental gap, optical gap and exciton binding energy, how each of them changes with the particle size, and how these changes are interrelated in one integrated study. Similarly, the exact nature of the delocalisation of the exciton and the frontier quasiparticle states, the states corresponding to the ionisation potential and electron affinity (Fig. 1), are not extensively explored.
Here we revisit the optical and electronic properties of hydrogen terminated SiNPs by performing GW-BSE and time-dependent DFT calculation on octahedral SiNPS containing 10 to 455 silicon atoms, see Fig. 1 and Fig. S1 (ESI†). We specifically focus on the (de)localisation of the electronic and optically excited state over the particles in both real and reciprocal space, and how the particle's fundamental gap, optical gap and exciton binding energy change with particle size. Finally, we also consider the effect of particle size on the predicted intensities of the excitations near the optical gap and predict the particles’ UV-Vis absorption spectra.
Subsequently, the quasiparticle spectrum and specifically the highest occupied and lowest unoccupied quasiparticle states, the ionisation potential and electron affinity, of the DFT optimised particles were predicted by different GW variants: single-shot G0W0, eigenvalue-only self-consistent GW (evGW) and quasiparticle self-consistent GW (qsGW) calculations.49–52 These GW calculations utilised the B3LYP orbitals as starting points and again used the def2-SVP or def2-TZVP basis sets. The results of the various GW calculations were then used as inputs for solving the Bethe–Salpeter equation to obtain vertical excitation energies, oscillator strength values, static polarizabilities and ultimately the nanoparticles’ optical gap values.53 For G0W0 the predicted properties will show a dependency on the functional used in the underlying DFT calculation. evGW and qsGW reduce and practically eliminate, respectively, this starting-point dependency by iterating the eigenvalues or the underlying ground state, respectively, until self-consistency is achieved.51 Moreover, in the case of finite-sized systems, the results of the combination of either evGW and qsGW and solving the Bethe–Salpeter equation agrees well with coupled-cluster benchmarks, as explicitly shown for singlet excitation energies for organic molecules,50 and yield excitation energies there that are clearly superior to G0W0-BSE.
Additionally, time-dependent density functional theory (TDDFT) calculations were carried out on the DFT optimised SiNPs, again using the same B3LYP functional and def2-SVP or def2-TZVP basis sets to predict the particle's absorption spectrum. These TDDFT calculations used the Tamm–Dancoff approximation to avoid TDDFT stability issues.54
Throughout all these calculations, version 7.5 of the Turbomole code was employed in combination with a tight integration grid (m5), stringent SCF convergence criteria (denconv set at 1 × 10−7) and the RI-J approximation.55–57 Additionally, the RI-K approximation was applied in all GW and BSE calculations, and in the G0W0 and evGW calculations we exploited the RIGW algorithms implemented in Turbomole,52 where only the highest occupied and lowest unoccupied quasiparticle states are explicitly calculated and the remainder of the Kohn–Sham spectrum is shifted accordingly. The latter approximation reduces the scaling of the calculations from N6 to N4, allowing for the study of much larger nanoparticles, but is not used for the qsGW calculations. The use of symmetry in the GW and BSE calculations in Turbomole is limited to Abelian point groups, therefore the GW and BSE calculations on SiNPs use the Abelian D2 or C1 sub-groups of the non-Abelian Td point group instead of the full Td symmetry of the particles. Finally, where plotting spectra or commenting on the strength of particular excitations we always use oscillator strength values calculated within the velocity gauge.
The character of the excited states predicted by the BSE and TDDFT was analysed in terms of the most prominent natural transition orbitals (NTOs),58 and using the excited state analysis software TheoDORE,59 in terms of the exciton size, charge-transfer character, and contributions of different atoms. Because TheoDORE is limited to analysing the results of calculation performed in C1, the latter analysis was limited in the case of BSE to particles for which we were able to perform BSE calculations in C1.
The effective static dielectric constants of the particles were estimated based on their static polarizability calculated from evGW-BSE. Firstly, the static evGW-BSE polarizability values α were converted into polarizability volume α′ values using:
α′ = α/(4pε0) | (1) |
requiv. = ((3/(4π))Vm)1/3 | (2) |
εr = (requiv.3 + 2α′)/(requiv.3 − α′) | (3) |
The bandlike nature of the frontier orbitals and NTOs was analysed by performing a discrete Fourier transform on the orbitals or NTOs. This Fourier transform was performed using an inhouse Python code based around routines from the numpy library and with a cube file of the orbitals/NTOS as input.61 The Fourier transform was plotted as a function of kx and ky defined as:
kx/y = 2π/lx/y | (4) |
The different properties of the SiNPs are plotted versus the radius of the silicon core of the SiNPs. The radius is calculated from the average edge length of the silicon cores (L), assuming the particles are ideal octahedra, as the distance between the centroid and the vertices:
R = (1/2)21/2L | (5) |
Finally, TDDFT optical absorption spectra were generated by representing all excitations in a given window above the lowest excited state by Gaussian functions with a width of 0.05 eV, followed by a point-by-point summation over these Gaussian functions for an equally spaced grid of points in that window. The size of the window depends on the size of the specific SiNP, as with increasing size of the SiNPs the spectrum of excited states becomes denser making it computationally harder to calculate all excitations in a window, as with the basis-set size.
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Fig. 2 Top: Quasiparticle spectrum for each SiNP showing the four highest occupied and four lowest unoccupied Kohn–Sham orbitals/quasiparticle states obtained from DFT, evGW and qsGW (the latter two where tractable). evGW fundamental gap values are also presented for each SiNP with the DFT Kohn–Sham gaps given in between parentheses. Bottom left: Plot of the optical gap values predicted using TDDFT, evGW-BSE and qsGW-BSE versus particle radius, as well fits to the evGW-BSE/def2-SVP and TDDFT/def2-SVP optical gaps where a is the optical gap of bulk silicon (1.1 eV), r is the SiNP radius, and b and n are fitting constants (see Table S10 for fitted values, ESI†), and data and fits for the experimental data of Wolkin et al.,63 and Furukawa et al.64 Bottom right: Plot of the exciton binding energy as a function of particle radius, as calculated from the fundamental gap and optical gap values predicted by evGW and evGW-BSE and DFT and TDDFT, respectively (open symbols for Si10H16 calculated using the lowest bright excited state). All results obtained using D2 symmetry, B3LYP and the def2-SVP basis-set, except if otherwise indicated. |
The results in the top panel of Fig. 2 are obtained using the def2-SVP basis-set. However, Tables S1–S6 (ESI†) show that values calculated with the larger def2-TZVP basis-set, calculations which, in the case of evGW and qsGW, are only computationally tractable for the smaller SiNPs, are very similar, and importantly show the same trend. The same holds for the results of qsGW calculations when compared to their evGW counterparts, validating the use of the RIGW algorithm which combined with the evGW scheme allows us to study much larger particles than possible otherwise. The DFT Kohn–Sham gap is considerably smaller than its evGW counterpart, while the energy of the highest occupied and lowest unoccupied Kohn–Sham states is less negative and more negative, respectively, than the corresponding evGW quasiparticle states. The fact that the DFT Kohn–Sham gap is smaller than the fundamental gap predicted by evGW, as well as the fact that DFT predicts a more negative energy of lowest unoccupied Kohn–Sham states than the evGW-EA, is expected because for pure density functionals, the unoccupied orbitals feel the same field of N − 1 electrons as the occupied orbitals instead of the correct N electrons.62 Introduction of exact exchange corrects this issue to a degree but not completely. However, importantly the DFT Kohn–Sham and evGW results display the same trend and the difference appears to reduce with increasing particle size.
For all method/basis-set combinations the fundamental gap is found, as expected from the analytical models discussed in the introduction and previous calculations,7–10 to vary as approximately one over the particle-size to some power n (see Fig. S2, ESI†). To exemplify this trend and to compare with the scaling of the fundamental gap previously reported in the literature and discussed in the introduction, we fit the fundamental gap to:
Δf = a + b/rn | (6) |
Finally, the highest occupied quasiparticle state for all octahedral SiNPs belongs to the T2 irrep and is hence triply degenerate. The lowest unoccupied quasiparticle state is non-degenerate and, in all cases, belongs to the A1 irrep. We do not observe a switch of the character of the highest occupied quasiparticle state for the smaller particles from T2 to T1 as seen in some early calculations on spherical cuts.9 The Kohn–Sham orbitals underlying the highest occupied and lowest unoccupied quasiparticle states for the three smallest SiNPs, Si10H16, Si35H36 and Si84H64 are shown in Fig. 3, where in the case of the triply degenerate highest occupied Kohn–Sham orbital, only one out of three is shown.
Similar to the fundamental gap, the bottom left panel of Fig. 2, as well Tables S7 and S8 (ESI†), show that def2-SVP and def2-TZVP predicted values agree reasonably well, with the def2-TZVP results shifted down relative to their def2-SVP counterparts, as do the results of BSE calculations on top of evGW and qsGW calculations. Moreover, in contrast to the fundamental gap, the evGW-BSE/qsGW-BSE and TDDFT predictions of the optical gap agree well, something that is not generally the case for inorganic nanoparticles, or at least without tuning the amount of exact exchange used in the TDDFT calculations.65,66 The optical gap of Si35H36 predicted by evGW-BSE/qsGW-BSE here agrees well with the onset of the BSE predicted spectra of Si35H36 by Rocca et al.67 Similarly, the TDDFT predicted optical gap value for Si35H36 is comparable to that predicted by Lethhonen and co-workers,14,15 and Wang and co-workers,12 both of which used B3LYP, and slightly smaller than that reported by Foerster and Besley,33 in line with the fact that the latter used CAM-B3YLP which is known to result in a blue-shift of the predicted optical absorption spectra relative to B3LYP.
Wolkin et al. reported the experimental photoluminescence energies (PLEs) of silicon nanoparticles of a range of sizes embedded in a porous silicon matrix generated through electrochemical and photo assisted stain etching, where the synthesized size range overlaps with that studied here by TDDFT/def2-SVP and evGW-BSE/def2-SVP.63 Similarly, Furukawa et al. reported the optical gap of hydrogenated silicon nanoparticles prepared by reactive etching with a similar size to that studied here.64 In both cases, the experimental values are ∼0.4–0.5 eV lower than the optical gaps predicted using TDDFT/def2-SVP and evGW-BSE/def2-SVP, (see Fig. 2 bottom left), but display a similar trend to the predicted optical gaps. In the Discussion section below, we discuss in detail how this discrepancy between the experimentally measured and predicted optical gap values is likely due to issues with the experimental data rather than the predictions.
For all method/basis-set combinations the optical gap is, like the fundamental gap, found to vary as approximately one over the particle-size to some power n. We fit the optical gap to a similar equation as the fundamental gap:
Δo = a + b/rn | (7) |
To provide a more qualitative picture, we predicted the static dielectric constant of the of the four smallest SiNPs: Si10H16, Si35H36, Si84H64, Si165H100 by calculating their static polarizability using evGW-BSE and using that as an input to obtain the predicted static dielectric constants. As can be seen from Fig. 4 the predicted static dielectric constants of the SiNPs are smaller than that of bulk silicon (11.4–12.0 experimentally,70,71 11.5–12.7 G0W0/G0W0 + BSE)72 but increase with particle size. The predicted values are similar and follow a similar trend with particle size to that predicted by Wang and Zunger using their empirical pseudopotential approach and those predicted by Delerue and co-workers using a tight-binding method.73,74
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Fig. 4 The static dielectric constant of the four smallest SiNPs Si10H16, Si35H36, Si84H64, Si165H100 as a function of particle size. The static polarizability was predicted using evGW-BSE, with D2 symmetry and B3-LYP, and used to obtain the static dielectric constant according to eqn (1)–(3). |
Overall, our predicted static dielectric constants and those from the literature, support the suggestion above that the larger exciton binding energy of the small SiNPs is the result of the reduced dielectric screening of the exciton in such particles, and the decrease in exciton binding energy with particle size is due to an increase in the static dielectric constant of the SiNPs tending towards the bulk value.
Finally, like the fundamental and optical gap, and as expected from the analytical models discussed in the introduction, the exciton binding energy varies as approximately one over the particle-size to some power n. To exemplify this trend, we fit the optical gap to a similar equation as the fundamental and optical gap:
EBE = a + b/rn | (8) |
Using TheoDORE it was possible to calculate the exciton radius, defined as the root-mean-square separation between the instantaneous electron and hole position,75 and the atoms over which the excited electron and hole are (de)localised for the lowest excited state as calculated with TDDFT and evGW-BSE. As TheoDORE is currently limited to analysing the results of C1 calculations, this limited us to an analysis of the smaller SiNPs, for which it was computationally tractable to run TDDFT and evGW-BSE calculations in C1, i.e., Si10H16–Si165H100 for TDDFT and Si10H16 and Si35H36 in the case of evGW-BSE. This analysis shows that the excited electron and hole are delocalised over the silicon atoms of the core with little or no contribution of the terminating hydrogens. The TheoDORE analysis also showed that the exciton radius was similar for TDDFT and evGW-BSE and increases proportionally with the SiNP radius (Fig. 5), as would be expected for an excited state delocalised over the whole nanocrystal volume. Visualisation of the hole and electron components of the leading natural transition orbitals for the lowest evGW-BSE excited state for the different particles, as shown in Fig. 5 (and Fig. S5 and S6 for their TDDFT counterparts, ESI†), yields the same picture.
Kocevski and co-workers have analysed the changing nature of the gap of SiNPs with particle size in terms of the Fourier transform of the frontier orbitals of spherical SiNPs.17 While bands strictly only exist under periodic boundary conditions and the wavevector k is only a proper quantum number in that limit, one can study signatures of ‘periodicity’ in the spatial nature of the frontier orbitals, and by extension the corresponding quasiparticle states, by considering the discrete Fourier transform of the orbitals as a function of kx, ky and kz. Fig. 7 shows the absolute values of the Fourier transforms of the three components of the Kohn–Sham HOMO and the LUMO as a function of kx and ky with the information along the z-direction projected on the xy plane by summing up the contributions along the z-axis for each xy value. The wavevectors are expressed as fractions of 2π/a0, where a0 is the lattice parameter of the primitive cell of bulk silicon (taken to be 3.89 Å). Just like Kocevski and co-workers we observe that with increasing particle size the spots in the plot become sharper, suggesting the orbitals become more band-like. Similarly, we also observe, just like them, that for the largest particles, the pattern in the spots becomes clearly different for the HOMO and LUMO in line with the valence band maximum and conduction band maximum for the bulk lying at different k values.
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Fig. 7 Fourier transforms of one of three degenerate HOMO orbitals (left) and the LUMO (right) of the smallest SiNP, Si10H16 (top) and the largest SiNP, Si455H195 (bottom), calculated using DFT and Td symmetry. The Fourier transforms of the other two HOMO orbitals can be found in the ESI† (Fig. S7). |
Interestingly, as can be seen from Fig. 7 and 8, the NTOs of the lowest excited state appear to display similar regular spot patterns in their Fourier transform as their frontier orbital equivalents. The NTOs, and thus the lowest energy exciton, appear to inherit the periodicity of the frontier orbitals, and thus the periodicity of the underlying atomic structure of the SiNP, as well as the differences in ‘localisation’ in reciprocal space between them.
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Fig. 8 Fourier transforms of the hole (left) and electron (right) component of the leading NTO of Si165H100, calculated using evGW-BSE and D2 symmetry. As the calculation were performed in D2 instead of the Td point group, the triply degenerate T2 excited state is described as a triplet of degenerate B1/2/3 excited states, the Fourier transforms of the leading NTOs of only one of those states are shown here. The others can be found in the ESI† (Fig. S8). |
Comparing the optical absorption spectra obtained with def2-SVP and def2-TZVP basis-sets for Si10H16 and Si35H36 we observe a small red-shift of the def2-TZVP results compared to their def2-SVP counterparts, in line with what was observed for the optical gap above. More interestingly perhaps, we also observe that the def2-TZVP predicted oscillator strength values are consistently larger than their def2-SVP counterparts, though importantly follow a similar trend.
As mentioned in the Results section, evGW-BSE, qsGW-BSE and TDDFT all predict very similar trends for the variation of the optical and fundamental gaps with particle size, where the optical and fundamental gap increase with decreasing particle size in 1/rn fashion, with similar n values for all methods and basis-sets. This 1/rn trend is, as discussed in the introduction, the expected trend from analytical models of the optical and fundamental gap of semiconductor particles, and is the trend seen in the experimental data for the optical gap (proxies) and previous calculations. However, the here predicted n values are smaller than those obtained from experiment and most of the previous calculations which report a n value or report sufficient data for us to fit a n value for. In the case of experiments, this difference in n is, as discussed above, probably related to issues with the interpretation of the experimental data. In the case of the comparison with the previous calculations, the calculations that give the larger n values use much more approximate methods than those used here, while the most recent calculation for both the optical and fundamental gaps give n values very similar to those obtained here.
As discussed above, based on the ev/qsGW-BSE predicted exciton binding energies, the excitation corresponding to the optical gap for all SiNPs studied here is clearly excitonic in nature with exciton binding energies one to two orders of magnitude larger than that for bulk silicon. The predicted exciton binding energies, >0.5 eV, lie in the range expected for Frenkel excitons. However, the fact that the exciton radii are similar to the SiNP radii and that the natural transition orbitals appear fully delocalised over the volume of the particles is more suggestive of a Wannier–Mott exciton. As is the fact that the Fourier transform of the natural transition orbitals show a similar spot pattern as the corresponding quasiparticle states, and thus likely display the same periodicity. The predicted large exciton binding energies thus seem completely driven by the poor dielectric screening by the vacuum outside the particles, in line with the fact that the reduction of the exciton binding energy goes hand in hand with an increase of the particle's predicted dielectric constant. The excitons in the SiNPs thus both combine Frenkel and Wannier–Mott like character.
Interestingly, the scaling of the predicted fundamental gap, optical gap and exciton binding energy with particle size is very similar in terms of the exponent n. This does not mean, however, that they vary by the same amount, as the b prefactors are significantly different. The variation in the evGW fundamental gap, our best estimate of the fundamental gap, with particle size, is approximately twice as large as the variation in the evGW-BSE optical gap. The exciton binding thus counter-acts the variation in the fundamental gap with particle size. This is expected based on the analytical models, discussed in the introduction, though such models would also predict that exciton binding energy, or at least its Coulombic component, would have a different scaling (a fundamentally different n value) than the fundamental gap, which is at odds with our predictions here. The similar scaling of the fundamental gap, optical gap and exciton binding energy with particle size also implies that these properties, by necessity, scale approximately linearly with each other. An example of that can be seen in Fig. S4 (ESI†) for the scaling of the evGW-BSE exciton binding energy with the evGW-BSE predicted fundamental gap.
In the literature the effect of nanostructuring on the intensity of the absorption of light near the optical gap is often discussed in terms of the ‘band gap’ changing from indirect to direct. However, our results here and the previous work of Kocevski and co-workers suggest that reducing the size of the particles does not so much change the ‘band gap’ from indirect to direct but rather that the ‘band structure’ becomes more smeared out, in line with the fact that k is not a good quantum number for a finite-size system. The reduction in oscillator strength with increasing particle size that we observe here, and which was previously observed by others,7,16 is probably driven simply by a reduction in the ‘overlap’ between the orbitals from which the electron gets excited and the orbital the electrons gets excited into. The ‘overlap’ in such a scenario then would be largest in the case of the smeared out ‘band structure’ for the small particles and reduces in size when with increasing particle size the orbitals, just like the valence band maximum and conduction band minimum for infinite crystals, become more sharply localised at different points in k-space.
The intensity of the excited state corresponding to the optical gap, and the excitations lying close to it in energy, are predicted to clearly increase with decreasing particle size. This increase in intensity upon reduction of the particle size has been explained in the literature as being due to a transition from an indirect to a direct ‘band gap’. However, Fourier transforms of the orbitals corresponding to the frontier quasiparticle states suggest that the origin more likely is the fact that the ‘band structure’ becomes smeared out with decreasing particle size, in line with the fact that the wavevector k is not a good quantum number for a finite-size system. This smearing increases the ‘overlap’ between the occupied and unoccupied states, and consequently also increases the oscillator strengths.
Fourier transforms of the natural transition orbitals that describe the excited states show that the lowest exciton inherits the periodicity of the frontier quasiparticle states. This combined with the delocalisation of the exciton over the volume of the particle and the large exciton binding energy means that the lowest exciton in these silicon nanoparticles combine aspects of Wannier–Mott (delocalisation and effect of periodicity of the underlying structure), and Frenkel (large exciton binding energy) excitons.
Footnote |
† Electronic supplementary information (ESI) available: DFT optimised structures of all relevant particles, tables of G0W0(-BSE), evGW(-BSE) and qsGW(-BSE) results and DFT predicted photoluminescence energies and Stokes shifts, and plots of the fundamental gap versus particle size, optical gap versus particle size using the lowest bright excited state for Si10H16 and of the natural transition orbitals obtained with TDDFT and additional Fourier transforms can be found in the supporting information. See DOI: https://doi.org/10.1039/d4cp00119b |
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