Metadynamical approach to the generation of amorphous structures: The case of $a$-Si:H

We present a dynamical approach to generate defect-free continuous-random-network (CRN) models of hydrogenated amorphous silicon ($a$-Si:H). Using the atomic coordination number of silicon as a collective variable and few configurational constraints, we have shown that classical metadynamics can be used to construct CRN models of $a$-Si with arbitrary concentrations of dangling-bond coordination defects. These defective networks have been subsequently hydrogenated to produce high-quality models of $a$-Si:H using ab initio total-energy calculations to generate hydrogen (H) microstructures for H concentrations from 7 to 22 at. %. The structural, electronic, optical, and vibrational properties of the models are examined, and the microstructure of the hydrogen distribution is analyzed and compared with experimental data from neutron scattering, spectroscopic ellipsometry, infrared spectroscopy, and nuclear magnetic resonance studies. The results obtained from the models are found to be in excellent agreement with the experimental data.

Figure 1

The free energy of $a$-Si from metadynamical simulations at 1000 K as a function of the collective variable $\xi$. The thin red curve indicates the region (of $\xi$) from where the configurations of $a$-Si have been sampled for hydrogenation.

Figure 2

Hydrogenation of a 3-fold dangling bond. A hydrogen atom (red) is placed on the surface of a central sphere (gray), whose center coincides with that of a defective Si atom (yellow), such that the Si-H bond retains the maximal tetrahedral character of the site.

Figure 3

Mass density of model networks of $a$-Si:H versus hydrogen concentration. Experimental data from Ref. [56] are included for comparison.

Figure 4

Neutron-weighted correlation functions (atoms/Å${}^2$) of $a$-Si:H from a 1000-Si-atom model (blue) with 22 at. % hydrogen and experiments. Experimental data (red) correspond to 22 at. % of hydrogen from Ref. [40].

Figure 5

Irreducible-ring distribution versus ring sizes for two models with different H concentrations. Hydrogenation causes a subtle change in the number of rings at a high concentration of hydrogen.

Figure 6

Tetrahedral units associated with ${\mathrm{Si}}_{\text{4-}n}{\mathrm{H}}_n$ configurations for $n=0$, 1, 2, 3. Silicon and hydrogen atoms are shown in yellow and white colors, respectively.

Figure 7

The volume distributions of individual ${\mathrm{Si}}_{4-n}{\mathrm{H}}_n$ tetrahedra for a model of $a$-Si (red) and $a$-Si:H (blue). The contributions from different H-bonded tetrahedra (see Fig. 6) are indicated. The vertical line (at 6.8 Å${}^3$) corresponds to the volume of an ideal ${\mathrm{Si}}_4$ tetrahedron in the crystalline-silicon environment.

Figure 8

Comparison of ${}^1\mathrm{H}-\mathrm{NMR}$ spectra from simulations (red and blue) and experiments (black). Simulated spectra correspond to $a$-Si:H models with 6.9 at. % (blue) and 14.7 at. % (red) of hydrogen. Experimental data are from Ref. [25] and correspond to a sample of $a$-Si:H with 10–15 at. % hydrogen.

Figure 9

Distribution of H clusters (white) in a 1000-Si-atom model with 17.9 at. % H. For visual clarity, individual clusters are enclosed within a hypothetical Gaussian isosurface (orange).

Figure 10

A few H clusters (white) in a 1000-Si-atom model with 10.6 at. % H. For visual clarity, a hypothetical Gaussian isosurface (orange) is constructed around each cluster.

Figure 11

Density of distributed H atoms (in percentage of total hydrogen) at low and high concentrations with a varying isolation radius. The hydrogen concentration of $a$-Si:H models is indicated in the plot.

Figure 12

An example of a distributed SiH configuration (red) in a 1000-Si-atom model of $a$-Si:H with 10.6 at. %. The radius of the spherical region shown above is 5.9 Å.

Figure 13

Density of clustered H atoms (in percentage of total H) as a function of total H concentration for a set of $a$-Si:H models. A hydrogen cluster is defined as a group of five or more H atoms in a spherical region of radius 3.8 Å.

Figure 14

Electronic density of states (DOS) of $a$-Si:H models in the vicinity of the band gap near the band edges for different hydrogen concentrations. The full valence and conduction bands are shown in the inset. The Fermi level is located at 0 eV.

Figure 15

A Tauc plot showing a linear extrapolation from the joint density of states (blue) to obtain an estimate of the optical gap (1.57 eV above) using a least-squares fit (red).

Figure 16

Variation of the optical gap (obtained from the Tauc plot) in $a$-Si:H with hydrogen concentrations. The widening of the gap with increasing hydrogen concentration is consistent with experimental data in Refs. [41, 67]. Experimental data shown above are from Ref. [41].

Figure 17

Dielectric function of $a$-Si:H from experiments (red) and simulations (blue) using the Tauc-Lorentz model. Experimental data are from Ref. [41] and correspond to a sample with approximately 11.6 at. % of hydrogen.

Figure 18

Atom-projected vibrational density of states of a 1000-Si-atom model of $a$-Si:H with 12.4 at. % H along with the inelastic neutron-scattering data (black) from Ref. [76]. The inset shows the calculated high-frequency vibrational modes and the experimental IR spectrum of $a$-Si:H from Ref. [21]. The lower panel shows the inverse participation ratio (IPR) of the vibrational eigenstates as a function of frequency.

Figure 19

A high-frequency stretching mode ($\nu =2182 {\mathrm{cm}}^{-1}$) associated with an isolated monohydride configuration in a model of $a$-Si:H with 12.4 at. % of hydrogen. The vibrational motion associated with this mode is found to be highly localized (with an IPR value of 0.95) and centered on the hydrogen (white) and silicon (yellow) atoms of the Si-H bond as indicated.