ANÁLISIS DE LA DEMANDA

After obtaining the a-SiO2 structures using ReaxFF, they were optimised further and

their electronic structures were extracted using DFT as described in chapter 2.3. To describe exchange and correlation, the non-local PBE0 TC LRC functional34_{was used}

along with the auxiliary density matrix method.73 _{This functional uses a truncated}

Coulomb operator to calculate the Hartree energy for which a truncation radius of 2.0 ˚

A was used. This functional was used as it contains a portion of Hartree-Fock exchange which improves the description of the band gap as well as describing localised states better than traditional local and gradient-corrected density approximations. Using

the Gaussian Plane waves method, the Gaussian basis set used was a double-ζ basis set with polarisation functions for both the Si and O atoms, while the plane wave basis set was truncated at 5440 eV (400 Ry). All of the systems were optimised individually using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. The structures were considered optimised when the forces on the atoms were converged to within 37 × 10−12 N (2.3 × 10−2 eV ˚A−1).

Figure 3.10: Histograms of the short-range geometrical properties of 320 DFT opti- mised models of a-SiO2. a) Si–O bond lengths. b) O–Si–O bond angles. c) Si–O–Si

bond angles.

The optimisation resulted in a slight change in the models’ atomic structures. As can be seen from Fig. 3.10, the Si–O bond lengths show similarly shaped distributions before and after optimisation, but the average bond length shifts up to 1.61 ˚A, in much closer agreement with the experimental value of 1.62 ˚A.74_{Similarly, the O–Si–O}

and O–Si–O bond angles, shown in Fig. 3.10, show similar shapes before and after optimisation. The average of the O–Si–O angle after optimisation is 109.47◦, very similar to the unoptimised structure. However, the O–Si–O angle averages at 146◦, very different from the unoptimised structure and much closer to experiment. The

total structure factor of the DFT optimised structures is plotted in Fig. 3.9 along with the original ReaxFF structures and experimental data. The structure factors are very similar at shorter ranges (< 7.5 ˚A−1) before and after DFT optimisation, with the optimised structures showing better agreement at longer ranges. The error compared to the experimental data is reduced after optimisation, indicating that the optimisation is a useful step in calculating the atomic structure of a-SiO2.

Figure 3.11: Histogram of the atomic displacements of the ReaxFF a-SiO2 models

before and after DFT optimisation.

To obtain a measure of the difference between the ReaxFF structures before and after DFT optimisation, their displacement fields have been calculated as:

D = [J3,1· (RDF T − RReaxF F)◦2]◦

1

2, (3.6)

where D is a matrix in which element i is the displacement of the ith _{atom, J} 3,1 is

matrices of the atomic coordinates before and after DFT optimisation, respectively. The displacements of all atoms are plotted in Fig. 3.11 for both Si and O atoms. The displacements peak at well below 0.1 ˚A and extend to 0.4 ˚A and 0.8 ˚A for O and Si atoms, respectively. These results indicate that the ReaxFF structures are very close to a DFT minimum, but that the parameters for the Si atoms could be better optimised.

Figure 3.12: Electronic densities of state from 320 models of a-SiO2. Each state is

broadened by a Gaussian with a sigma value of 0.1 eV. The black curve is the total, while the red and blue curves are the densities of state projected onto Si and O atoms, respectively. The areas of the curve which are filled in to the baseline indicate occupied states, while the curves which are not filled in indicate unoccupied states. The states’ energies are normalised so that the Fermi level lies at 0.0 eV.

To assess the electronic structures of the a-SiO2 models, their electronic densities

of state were calculated. Fig. 3.12 shows a summation of these extracted densities of state from the 320 models of a-SiO2. Each state is broadened by a Gaussian

with a sigma value of 0.1 eV to simulate homogeneous broadening. In addition, the densities of state were projected onto Si and O atoms in order to assess each

atom type’s contribution to the total density of states. All states are aligned so that the Fermi level lies at 0.0 eV and the occupied states lying below the Fermi level are filled in to the base line. Immediately, one can see that a-SiO2 is an insulator

with a very wide band gap. The average band gap extracted from these calculations is 8.1 eV and ranges from 7.1 to 8.4 eV. This is a slight underestimation of the experimental SiO2 band gap, which has been measured optically as 8.8 eV.75 The

traditional exchange-correlation functionals, such as the local density and gradient corrected approximations, are known to underestimate the band gap of materials, with the band gaps of these models calculated with PBE (a gradient corrected functional) averaging at 6.4 eV . However, the inclusion of Hartree-Fock exchange is known to improve the calculated band gap. Increasing the amount of Hartree-Fock exchange increases the band gap, but it also changes the chemical properties of the material in an unpredictable manner. In fact, in the literature many studies have used the proportion of Hartree-Fock exchange as an effective tuning parameter to replicate the band gap. However, in this thesis the amount of HF exchange will remain at 0.2, the value quoted in the original PBE0 TC LRC functional’s paper. It is also clear to see that the top of the a-SiO2 valence band is dominated by contributions

from O atoms. In fact, these are contributions from O non-bonding ‘p’ states. The bottom of the conduction band is made up of hybrid Si ‘sp’ states as well as non- bonding O ‘p’ orbitals as well. An interesting property that emerges from these calculations is that the extrema of the bands are not entirely delocalised, but are instead partially localised at particular locations in these structures. The disorder of the amorphous SiO2 structures leads to deviation from band theory, with the states

induced by disorder known as the band’s Urbach tails in the literature.76 The ability of these Urbach states to act as precursors to strong localisation is known as Anderson localisation 77 _{and these states in a-SiO}

3.3.3

Summary

A molecular dynamics procedure is established to generate a-SiO2 structures by ad-

justing the time step, thermostat and barostat and following a melt and quench pro- cedure. The ReaxFF potential is used to generate 320 a-SiO2 samples. The resulting

amorphous structures are characterised and found to agree very well with experiment; the density is very close to the experimental value as are the total structure factors. These structures were then optimised using DFT leading to an even better agreement of the atomic structure with experiment. Their electronic structures were extracted showing that the calculated band gaps are in reasonable agreement with experiment, albeit a little too low. These results indicate that the use of molecular dynamics and ReaxFF can provide an accurate description of the structure and disordered nature of amorphous silica. Further optimisation using DFT is not necessary to improve the description of the atomistic structure, but is necessary to extract the electronic structure.