The Effect of Particle Size on the Optical and Electronic Properties of Magnesium Oxide Nanoparticles

The (band) edge states, fundamental gaps, optical gaps, exciton binding energies and UV-Vis spectra for a series of cuboidal nanoparticles of the prototypical oxide magnesium oxide (MgO), the largest of with has 216 atoms and edges of 1 nm, were predicted using many-body perturbation theory (ev GW -BSE). The evolution of the properties with particle size was explicitly studied. It was found that while the edge states and fundamental gap change with particle size, the optical gap remains essentially fixed for all but the smallest nanoparticles, in line with what was previously observed experimentally. The explanation for these observations is demonstrated to be that while the optical gap is associated with an exciton that is highly localised around the particle’s corner atoms, the edge states, while primarily localised on the magnesium corner atoms (electron) and oxygen corner atoms (hole), show significant delocalisation along the edges. The strong localisation of the exciton associated with the optical gap on the corner atoms is argued to also explain why the nanoparticles have a much smaller optical gaps and red-shifted spectra than bulk MgO. Finally, it is discussed how this non-quantum confinement behaviour, where the properties of the nanoparticles arise from surface defects rather than differences in localisation of edge or exciton states, appears typical of alkaline earth oxide nanoparticles, and that the true optical gap of bulk crystals of such materials is also probably the result of surface defects, even if unobservable experimentally. states, fundamental gap, optical gap and exciton binding energy values predicted by BSE/ G 0 W 0 (AC)/def2-SVP and BSE/ev GW (AC)/def2-SVP for a series of cubic and MgO nanoparticles. family,


Introduction
The effect of going from the bulk to nanosized particles on the optical and electronic properties of materials is interesting both from a fundamental and applied perspective. 1 Practically it offers an alternative way of tuning material properties besides changing composition while conceptually it raises questions about what exactly results in the properties of nanoparticles differing from those of the bulk. Quantum confinement, [2][3] where the size of the particle constrains the size of the exciton --the excited-electron hole pair formed by the absorption of light--to a size smaller than in the bulk, is often invoked but requires the states involved to be delocalised over the particle, which is unlikely to be the case for more ionic materials. Quantum confinement also only explains blue-shifts with respect to the bulk, as e.g. observed for CdS, 4 CdSe, 5-6 PbS 7-8 and PBSe 9 nanoparticles, but not red-shifts, as e.g. observed for MgO, [10][11] CaO 11 and SrO 12 particles. An alternative mechanism by which the properties of nanoparticles may differ from the bulk involves the localisation of relevant states on low-coordinated surface atoms, which are ubiquitous on (small) nanoparticles. While experimental spectroscopy can clearly demonstrate the effect of nanostructuring on the electronic and optical properties, elucidating the atomic scale origin of these changes requires the combination of experiment with theory.
Magnesium oxide (MgO) nanoparticles are an ideal system to study differences in the optical and electronic properties between nanoparticles and the bulk and to synthesize the results of experiment and theory. Well-defined MgO nanoparticles as small as 3 nm have been prepared experimentally by means of chemical vapour deposition without the need for capping agents and found to display UV-Vis diffuse reflection and photoluminescence spectra that are significantly red-shifted relative to bulk magnesium oxide. [10][11] The lowest-energy most red-shifted exciton peak in the reflection spectrum of bulk MgO, [13][14] the bulk optical gap, is located at 7.7 eV (161 nm) while the lowest peak in the reflection spectrum of the 3 nm particles lies at ~4.6 eV (270 nm). 10 These 3 nm nanoparticles are confirmed to have the same rocksalt structure as bulk magnesium oxide, ruling out that the observed red shift is the result of major structural changes. Indeed, computational global optimisation studies show that in contrast to materials that in the bulk crystallise with the zincblende or wurtzite structure, e.g. zinc or cadmium sulfide, [15][16][17][18][19] the lowest energy nanoparticles of materials that crystallise with the rocksalt structure, such as MgO, [20][21][22] are, even in the absence of capping agents, generally cuts from the that crystal structure. The fact that magnesium and oxygen are relatively light elements also means that relativistic effects including spin-orbit coupling will be small in MgO nanoparticles in contrast to other rocksalt nanoparticles, such as those made from PbS or PbSe. Interestingly, while the MgO nanoparticles' UV-Vis spectra are significantly red-shifted with respect to the bulk, the largest change in the diffuse reflectance spectra when changing the average particle size from 3 to 10 nm is not a shift in the peak positions but their relative intensities. 10 In contrast to the case of the optical properties, there appear to be no reports on the electronic structure of MgO nanoparticles: i.e. the fundamental gap, the energy required to generate a non-interacting excited electron and hole pair rather than an interacting exciton and/or the energies of the occupied (the negative of the ionisation potential, -IP) and unoccupied (the negative of the electron affinity, -EA) edge states (see Fig. 1). The latter concepts in the case of periodic crystals, where it is appropriate to speak of bands, map on to the bandgap, valence band maximum and conduction band minimum, respectively. There are some reports in the literature of the fundamental gap narrowing for thin films of MgO. [23][24] However, as this is generally observed by means of electron energy loss spectroscopy (EELS) and as one of the loss mechanisms in EELS involves the generation of excitons, the feature at ~6 eV in the EELS spectrum of thin magnesium oxide films linked to the apparent narrowing of the fundamental gap in reality might be due to surface exciton formation 25 and thus be evidence of the narrowing of the thin films' optical gap instead.
The optical properties of MgO nanoparticles were previously studied 26-30 using time-dependent density functional theory (TD-DFT). The results of these TD-DFT calculations for whole nanoparticles was found to be rather sensitive to the exact density functional used 30 because of the well-known issue 31-32 of TD-DFT where chargetransfer excitations are spuriously stabilised with respect to local, i.e. non-chargetransfer, excitations. Well-chosen density functionals with the optimal amount of exact exchange can reproduce the key features of the experimental spectra of magnesium oxide nanoparticles: the red shift with respect to the bulk and the intensity change for the diffuse reflection peaks when changing particle size, 30 but the strong dependency on the amount of exact exchange makes these calculations more empirical than desirable. TD-DFT calculations using embedded cluster calculations [26][27][28][29] where only a region, e.g. a corner, of the nanoparticle is describe explicitly using TD-DFT and the effect of the rest of the particle on this region is described in terms of classical point charges, appear to be less sensitive to the functional choice, perhaps because many potential charge-transfer excitations, e.g. from oxygen corner atoms to magnesium corner atoms, are by definition absent in embedded cluster models. 30 It is however difficult to explicitly study the effect of particle size in embedded cluster calculations.
An additional complication when using TD-DFT is that the optical gap and the UV-Vis spectra in general are not treated on the same footing as the particle's electronic properties. The edge states of a particle and its fundamental gap can in principle be calculated within the framework of ground state density functional theory (DFT) in two different ways, from the Kohn-Sham (KS) orbital energies or using DDFT. In the former case, the occupied and uncopied edge states map on to the highest occupied and lowest unoccupied KS orbitals and the fundamental gap on to the energy gap between these two orbitals, the KS gap. In the latter case the energy of the occupied edge state is calculated from the difference in total energy between the neutral particle and the particle with one electron less, that of the unoccupied edge state from the total energy difference between the neutral particle and the particle with an extra electron, and the fundamental gap from the difference in energy between the unoccupied and occupied edge states. The DDFT approach is preferred as it can be shown that the KS gap, at least for pure density functionals, behaves more like the optical than the fundamental gap because the KS unoccupied orbitals feel the same field of N-1 electrons as the KS occupied orbitals instead of the correct N electrons. 33 However, regardless, it is not a given that the optimal density functional for calculating the spectra and optical gap values of particles is also optimal for calculating the particles' electronic properties. This is especially critical for properties such as the exciton binding energy, the difference between the fundamental and optical gap and a measure of how strongly excitons are bound (see Fig. 1), which span both worlds. DDFT predictions of edge state energies of MgO particles, including particles with defects or in the presence of grain boundaries, have previously been reported by Shluger and co-workers. [34][35][36]

Methodology
The geometry of the nanoparticles was optimised in DFT calculations, using the  32 .

(MgO) 4 and (MgO)
First, the effect of the different GW approximations and implementations is studied by performing calculations on the smallest two perfect cubes, (MgO) 4 and (MgO) 32 . Tables   1 and 2 give the edge states, fundamental gap, optical gap and exciton binding energy values predicted by the different method combinations for the two particles.
Concentrating first on the effect of basis-set, with increasing basis-set size the edge states both move to deeper, more negative, values while the fundamental and optical gap increase and the exciton binding energy more or less remains the same. Generally, as would be expected, the shift when going from the triple-zeta def2-TZVPP to the quadruple-zeta def2-QZVPP basis-set is smaller than when going from def2-SVP to def2-TZVPP, except for the unoccupied edge state, for which the shift is similar. A calculation using the augmented Dunning aug-cc-pVTZ basis-set, suggest that the effect of adding additional diffuse basis functions is minor. Table 1 Occupied (-IP) and unoccupied (-EA) edge states, fundamental gap (D F ), optical gap (D O ) and exciton binding energy (EBE) values of (MgO) 4 as calculated by the different method combinations (TZVPP=def2-TZVPP, accTZ=aug-cc-pVTZ, QZVPP=def2-QZVPP, SVP=def2-SVP).  It should be noted here that both the GW and LR-CCSD calculations ignore vibronic and zero-point motion effects and that as such this is a fair comparison but that the true experimental gap likely will be smaller as a result of such effects. Change in edge state positions and optical gap with particle size. Table 3  probably explains the absence of variation in optical gap with particle size. In contrast to the optical gap, the edge states and by extension also the fundamental gap do change with particle size, even if the picture is confusing.  While the lowest excitation corresponding to the optical gap for all particles involves, as discussed above, the 3-coordinated oxygen corner atoms and the immediately adjacent 4-coordinated magnesium edge atoms, the ground-state Kohn-Sham orbitals from DFT corresponding to the occupied and unoccupied edge states for all cubic and cuboid particles considered are localised on the 3-coordinated oxygen atoms, with minor contributions of the oxygen atoms on the edge or diagonal between these 3-coordinated oxygen atoms, and 3-coordinated magnesium atoms, respectively (see Fig. 3B). The fact that in the lowest excitation the electron gets excited from 4-coordinated magnesium edge atoms adjacent to the 3-coordinated oxygen corner atoms rather than from the 3-coordinated magnesium corner atoms combined with the fact that the optical gap is much smaller than the fundamental gap for all particles suggests that this excitation is excitonic in character. Cuboid MgO particles with odd-numbered faces. Table 4  can be found in Table S2 in the supplementary material. In the previous (TD-)DFT study of MgO nanoparticles it was noted that these odd-numbered faces behaved as if polar, even if extended (100) surfaces are not polar in the conventional sense. 30 Because one odd-membered face has one more magnesium atom than the opposite other odd-membered face and the latter has one more oxygen atom than the former, While particles with odd-membered faces larger than (MgO) 27 were not considered here, the data suggests that for very large particle dimension the fundamental and optical gaps of such particles might go to zero with the particles becoming metallic. However, as discussed below this situation is unlikely to be ever encountered in experiment.  Compared to the previous TD-DFT calculations 30 using the range-separated CAM-B3LYP functional the main difference is the predicted character of the excitations responsible for the 240 nm shoulder. TD-CAM-BLYP predicts that one of the excitations contributing to that shoulder corresponds to an excitation of an electron from a 3coordinated corner oxygen atom to a 3-coordinated corner magnesium atom, something that is not observed in the BSE/evGW predicted spectra to much higher excitation energies / shorter wavelengths. were calculated using BSE/G 0 W 0 and BSE/evGW, see Table 5. As can be seen from Table 5 As discussed above the edge states and the fundamental gap in contrast to the optical gap do change with particle size. The relevant orbitals are again relatively localised but more delocalised than the excited state associated with the optical gap.
The origin of the variation of these electronic properties with particle size is unknown.
Perhaps more interestingly is the fact that the nature of the variation is unexpected.
The occupied edge state is found to move to shallower values, the unoccupied edge to deeper values and hence the fundamental gap decreases with particle size. It turns out to be impossible to predict the (MgO) 108 spectrum using BSE/evGW, even when using the def2-SVP basis-set, due to the very large memory and disk-space requirements to calculate more than just the lowest excited states for this particle.
However, it stands to reason that the observed trend in the intensities of the peaks in the experimental reflection spectra, where going from 3 to 10 nm particles the relative intensity of the 270 and 240 nm shoulders/peaks decreases and that of 220 nm peak increases, finds its origin in the fact that the most red-shifted peaks involve excitations localised on exclusively 3-coordinated corner and 4-coordinated edge atoms while the 220 nm peak also involve 5-coordinated terrace and 6-coordinated bulk atoms and that when increasing the particle size the number of corner and edge atoms relative to surface and bulk atoms decreases. For the same reason and in analogy to the discussion above for the fundamental gap, even for macroscopically large MgO particles the true optical gap would likely be the same as that predicted here for nanoparticles but the optical gap observed in experiment would probably be the bulk value.
The exciton binding energy decreases with particle size, driven by the reduction in the fundamental gap with particle size. In the case of pure cubes extrapolation to the infinite particle limit by plotting the BSE/evGW/def2-SVP exciton binding energy of the lowest energy exciton of the three cubes against the inverse of their edge lengths yields a value of 2.4 eV. This value is more than an order of magnitude larger than the binding energy of the n=1 bulk exciton, 80 meV, [63][64] which is in line with the former probably 65 being a Wannier-Mott exciton while the lowest exciton in the nanoparticles, as discussed, is much more localised and hence more like a Frenkel exciton. The nanoparticles with odd-membered faces have rather different optical and electronic properties relative to their counterparts that only have even-membered faces.
However, particles displaying such odd-membered surfaces are unlikely to be observed in experiment as the same ground-state dipole moment that gives rise to these different optical and electronic properties also destabilises them energetically. Previous global optimisation studies find that while for (MgO) 18 the cuboidal rocksalt structure is still the predicted global minimum, as it is for particles with only even-membered faces (i.e. (MgO) 24 , (MgO) 32 and (MgO) 40 ), for (MgO) 27 a reconstructed cuboidal structure with a much reduce ground-state dipole moment (0.3 instead of 32.6 Debye) is found to be more stable 21 (see Fig. 6). As expected this reconstructed (MgO) 27 particle also has a larger optical gap (4.48 eV instead of 3.95 eV when calculated with BSE/evGW(AC)/def2-SVP, see table S3 in the supporting information), which is similar to that of the cubic particles with even-membered faces. In analogy to classically polar surfaces, these particles with odd-membered surfaces can thus be thought to reconstruct to eliminate the large dipole moment and in the process also eliminate their more out of kilter optical properties. particles with odd membered faces is predicted to be essentially the same and to not vary with the particle size. By extrapolation it can be assumed that the same holds true for macroscopic MgO particles, even if this might be hard to observe experimentally due to the much smaller surface to volume ratio for such particles. The BSE/evGW spectrum predicted for (MgO) 32 agrees well with the experimental spectra for MgO nanoparticles reported in the literature.
Nanoparticles with odd membered faces are predicted to have reduced optical gaps and display large ground state dipole moments perpendicular to the odd membered faces.
The presence of such large dipole moments destabilises these particles relative to particles that lack odd membered faces and hence can be argued to reconstruct to reduce the dipole moments, akin to what happens in the case of polar surfaces.