\section{Introduction}
The structure of amorphous silicon ({{\it a}-Si}) is well
represented by the continuous random network (CRN)
model of Zachariasen~\cite{Zach1932}. The CRN model
of {{\it a}-Si} suggests that each atom is bonded to
four neighboring Si atoms, which form an approximate
tetrahedral atomic arrangement in the amorphous environment.
The network is topologically distinct from its crystalline
counterpart ({{\it c}-Si}) owing to the presence of 5-member
and 7-member rings. In addition, a considerable number
of hexagonal rings and a few higher-member rings are also
present in the amorphous network. The
pair-correlation function (PCF) of {{\it a}-Si} obtained from
CRN models indicates that radial correlations
typically extend up to a distance of 15 {\AA}. Although
the actual structure of laboratory-grown samples of
{{\it a}-Si} may differ from this simple CRN picture, except
for a few properties, the CRN model provides an overall
good description of structural, electronic, and
vibrational properties of {{\it a}-Si} that mostly rely on
the short-range order ($\approx$ 5 {\AA}) and, to a
lesser extent, the medium-range order ($\approx$ 5--20
{\AA}) of the network.

Although the structure of {{\it a}-Si} has been extensively studied
by using computer-generated models on the radial
length scale of 10--15 {\AA}, there exist only
a few studies~\cite{Elliott:1991, Biswas:2020, Biswas:2015, Dahal:2019} that
discuss the network structure of {{\it a}-Si} on the
medium-range length scale of 20 {\AA} and
beyond. This is partly due to the fact that structural
and electronic properties of {{\it a}-Si} are generally found to be not
particularly dependent on the medium-range structure
beyond 15 {\AA} and in part to the computational
complexity of conducting quantum-mechanical calculations,
using density-functional theory (DFT), for large
models.  However, this observation does not necessarily
imply that no medium-range structure exists in
{{\it a}-Si}~\cite{Gibson:1997}.  In this paper, we address
this aspect of the problem by studying the network structure
of {{\it a}-Si} using atomistic models of sizes 21,952 and
400,000 atoms. In particular, we examine two important
aspects of the medium- and extended-range structures
of {{\it a}-Si} that have been sparsely reported in the literature.
The first problem involves the presence of weak but
noticeable radial oscillations in the PCF at
distances of 20--40 {\AA}.
This was first reported by Uhlherr and Elliott~\cite{Uhlherr:1994}
and it was given the name extended-range oscillations
in the PCF of {{\it a}-Si}. The second issue is directly related
to the first and it concerns the effect of the medium-range
order beyond 15 {\AA}, and possibly the extended-range
order, on the first sharp diffraction peak (FSDP)
of {{\it a}-Si}. The latter corresponds to the first peak of
the static structure factor~\cite{Xie:2013},
$S(Q)$, at $Q$ = 1.99 {\AA}$^{-1}$ in {{\it a}-Si}. In the following, we
use the term medium-range order (MRO) to imply
ordering on the length scale of 5--20 {\AA}, whereas the
term ERO indicates structural ordering beyond 20 {\AA},
including extended-range oscillations.

The role of the medium-range order (MRO) in amorphous networks
has been studied extensively in an effort to understand
structure-property relationships in network-forming
glasses, for example,
oxides~\cite{Aniya:2004,Shyam:2016,Mei:2008,Salmon:2006,Sampath:2003}
and chalcogenides~\cite{Vashishta:1989, Phillips:1981,
Lucovsky:1987,Iyetomi:1991,Armand:1992}. The MRO in these
systems typically manifests itself as the FSDP, and the position,
width, and intensity of the FSDP characterize the
length scale associated with the MRO.
The results from numerous experimental~\cite{Sheng:2006, Yang:2021, Ma:2009, Cormier:1998, Salmon:2007, Mei:2008, Hazra:2004}
and computational studies~\cite{Elliott:1991, Hirata:2011, Sharma:2006, Tanaka:1998, Nishio:2013, Du:2006, Deringer:2021}
indicate that the MRO/ERO in glassy systems can extend up
to a distance of 30 {\AA} and that it can play an
important role in determining a number of materials
properties of network-forming glasses. By contrast, results
for tetrahedrally-bonded elemental amorphous semiconductors,
such as {{\it a}-Si} and {{\it a}-Ge}, are
few and far between. Uhlherr and Elliott~\cite{Uhlherr:1994}
studied the presence of extended-range oscillations in
{{\it a}-Si} by analyzing experimental neutron-diffraction data
of Fortner and Lannin~\cite{Fortner:1989} and the
pair-correlation data obtained from atomistic models
of size 13,824 atoms.~\cite{Holender:1991} The authors
concluded, via the Fourier inversion of the structure
factor in the vicinity of the FSDP region, that the
radial oscillations can extend to at least 35 {\AA} and
that it arises from the propagation of second-neighbor
radial atomic correlations. Recently, Roorda et al.~\cite{Roorda:2012}
reported the presence of ERO in amorphous Si/Ge
using x-ray diffraction measurements at high
resolution. The PCF obtained in their study from
the Fourier transform of diffraction data shows the
presence of ERO beyond 20 {\AA} in both {{\it a}-Si}
and {{\it a}-Ge} samples.  The authors also noted
that the (spatial) periodicity and decay length of
the MRO/ERO increase upon thermal annealing.
In view of these observations, the main task of the
present study is to examine the presence of the ERO
in large realistic models of {{\it a}-Si} by a direct analysis of the
pair-correlation function and their partial
counterparts associated with distant coordination
shells of amorphous silicon.

The rest of the paper is arranged as follows. In
section II, we have provided a description of
the computational methods employed here to generate
atomistic models of {{\it a}-Si} and a set of partially-ordered
networks of Si atoms. This is followed by results
and discussion in section III. The origin of the
ERO is addressed from a real-space point of view
of the network structure of amorphous silicon.
The relation between the ERO and structure of the
FSDP is also examined in this section by constructing
a semi-empirical expression for the structure factor
of {{\it a}-Si} in the Gaussian approximation. This is
followed by conclusions of our work in section IV.

\section{Computational method}
The present study involves the use of three different
sets of models. The first set consists of {{\it a}-Si} models
obtained from using the Wooten-Winer-Weaire (WWW)~\cite{Wooten:1985,Barkema:2000}
algorithm. The second set comprises {{\it a}-Si}
models produced from large-scale molecular-dynamics
(MD) simulations.
The third set includes three different types
of partially-ordered networks of Si atoms, denoted by
M1, M2, and M3. These are not realistic models of
{{\it a}-Si} but have varying degrees of radial
ordering in the respective PCF up to a radial length
of 6 {\AA}. In order for the ERO
to manifest in the PCF of {{\it a}-Si} at radial distances
of 20--40 {\AA}, it is necessary for the models to be
sufficiently large, consisting of a few tens to several
tens of thousands of atoms. To this end, the sizes
of the models were chosen to be 21,952 atoms and 400,000
atoms, which suffice to establish an unambiguous
presence of the ERO in the PCF.  In this study, we
generated a set of three independent M1/M2/M3 and
WWW models and three MD models for the purpose of
configurational averaging of data.

The MD models were produced by initially placing 400,000 Si atoms
randomly in a cubic simulation cell of length 202.4 {\AA},
so that no two atoms could be at a distance of less
than 2 {\AA}.  The mass density of the models corresponds to
2.26 g.cm$^{-3}$, which is close to the
experimental value~\cite{Custer:1994, Laaziri:1999} of the
{{\it a}-Si} density, 2.25--2.28 g.cm$^{-3}$, depending upon the method
of preparation and experimental conditions. The modified
Stillinger-Weber potential~\cite{Vink:2001,SW:1985} was
used to calculate the total energy and forces and the
velocity-Verlet algorithm was employed to integrate the
equations of motion in canonical ensembles. The initial
temperature was set at 1800 K and the system was equilibrated
for 20 ps at 1800 K. The temperature was then gradually decreased, by using
a chain of Nos\'{e}-Hoover thermostats~\cite{Nose:1984, Hoover:1985},
from 1800 K to 300 K at an average cooling rate of
$5\times10^{12}$ K/s.  The final
structures from the MD simulations were further subjected
to geometry optimization using the limited-memory BFGS
algorithm, as described by~\citet{Ray:2018}.  Atomic
configurations
were collected during the course of simulations once
the configurations satisfied a set of convergence properties,
involving a minimum value of the width of the bond-angle
distribution and the number of 4-fold-coordinated
atoms in the network.

The second set of models were produced by using the
WWW method. Here, we employed the modified version
of the algorithm, developed by Barkema and
Mousseau~\cite{Barkema:2000}. The method
essentially consists of the following steps: 1)
Generate a random configuration and construct a
neighbor list of atoms using an appropriate cutoff
value, such that the network is tetravalent as far
as the list is concerned; 2) Employ the WWW bond-switching
algorithm~\cite{Wooten:1985, Barkema:2000} to produce
a new configuration and accept or reject the configuration
upon local relaxation of the network via the Monte
Carlo method. The bond-switching procedure largely
maintains the tetravalent character of the atomic
network during simulations, and local relaxations
were performed by using the nearest-neighbor-based Keating
potential~\cite{Keating:1966}; 3) Relax the resulting
configuration from step 2 at a regular but infrequent
interval to include the structural information from
beyond the first shell of neighbors, by using a
generalization of Weber's adiabatic bond-charge
model~\cite{Weber:1977}. For a description
of the method, see \citet{Barkema:2000}.

In addition to the WWW and MD models of {{\it a}-Si}, we have
also generated a set of disordered networks, M1--M3,
of Si atoms. These networks are partially ordered and they can be
classified by the degree of radial correlations present
in the respective PCF. Specifically, M1 models are
highly disordered and have very little or no radial
correlations in the PCF. By contrast, M2 models are
characterized by the presence of a well-defined
first peak and radial correlations up to 3 {\AA}.
Likewise, M3 models exhibit radial correlations
up to 6 {\AA} with a pristine first peak and a
part of the second peak, with a well-defined gap
between the peaks. The M2 and M3 models were
generated by adding one atom at a time in the
simulation cell so that the addition of each atom
satisfied a set of geometric constraints in order
to produce radial correlations up to a length
of 4 {\AA} and 6 {\AA}, respectively. The sizes
of the WWW and M1/M2/M3 models were chosen to be
21,952 atoms, with a cubic supercell of linear size
77.03 {\AA}.

Apart from the WWW, MD, and M1 to M3 models, we have
also employed a number of disordered amorphous
silicon ({\it{da}}-Si) and disordered crystalline silicon ({\it{dc}}-Si)
configurations in this study. These configurations
were produced by including structural disorder in
pristine {{\it a}-Si} and diamond {{\it c}-Si} structures
via random displacements of atoms, using $r_{i, \alpha} \to
r_{i, \alpha} + \sigma\,p_{i,\alpha}$, from their
original positions. Here, $r_{i,\alpha}$ is the
$\alpha$-th component ($\alpha = x/y/z$) of the
atomic position at site $i$, $\sigma$ is the maximum
value of the atomic displacement in {\AA}, and
$p_{i,\alpha}$ is a random number, which is uniformly
distributed between -1 and +1. The values of $\sigma$
were chosen from 0.2 {\AA} to 1.2 {\AA},
which correspond to a distortion of the Si--Si bond
length by 8--51\% from its average/ideal
value of 2.36 {\AA} in {{\it a}-Si}/{{\it c}-Si}. It may be noted that
a value of $\sigma$ of the order of 0.3 {\AA}
satisfies the Lindemann's criterion of melting,
producing liquid-like structures of {{\it a}-Si} and
{{\it c}-Si}. Thus, the {{\it dc}-Si} configurations with $\sigma
\gg$ 0.3 {\AA} are considerably disordered
compared to their counterparts with $\sigma
\le $ 0.3 {\AA}.

Given a distribution of atoms in a disordered
network, the structure factor can be obtained
from the Fourier transform of the reduced
PCF, $G(r)$. Assuming that the distribution of atoms
in the network is homogeneous and isotropic,
the structure factor, $S(Q)$, is given by,
\begin{eqnarray}
S(Q) & = & 1 + \frac{4\pi n_0}{Q} \int_0^{\infty} r[g(r)-1] \sin(Qr)\, dr \notag \\
     & \approx & 1 + \frac{1}{Q} \int_0^{R_c}  G(r) \sin(Qr)\, dr,
\label{eq1}
\end{eqnarray}
where $g(r)$ is the conventional pair-correlation function
(PCF), $G(r)= 4\pi n_0 \, r\,[g(r)-1]$ is known as the
reduced PCF, and $n_{0}$ is the average number density
of the system. For finite-size models, the upper limit
of the integral can be replaced by $R_c = L/2$ by
using the periodic boundary conditions, provided
$g(r) \rightarrow 1$ as $r \to R_c$. We shall see later that
this condition is amply satisfied by models for which
$R_c$ is of the order of 20 {\AA}.

\section{Results and Discussion}
\subsection{Extended-range oscillations in the PCF of {{\it a}-Si} }
We begin by establishing the unambiguous presence of
radial oscillations in the PCF of {{\it a}-Si} at a distance
of 20--40 {\AA}. Since the calculation of the PCF
beyond 20 {\AA} requires sufficiently large models
of {{\it a}-Si}, we first examine the large MD models,
consisting of 400,000 atoms. Thereafter, we proceed to
determine the origin of these oscillations by
analyzing the three-dimensional network
structure of these 400,000-atom models and a set of
21,952-atom models obtained from the WWW method.
The results from these models will be compared with
the same from the partially-ordered networks, M1 to
M3, having varying degrees of radial ordering up
to a distance of 6 {\AA}.
The PCFs of the partially-ordered networks, from M1
to M3, are shown in Fig.\,\ref{rdf_M}, along with
the results from the 21,952-atom WWW models of
{{\it a}-Si}. It is evident from the plots that the M2
and M3 models show radial correlations of
up to 4 {\AA} and 6 {\AA}, respectively. The
M1 models, on the other hand, exhibit small
radial correlations up to 3 {\AA}, which mostly
originate from the imposed constraint of a minimum
separation distance of 2 {\AA} between any two
atoms in the network.

Figure \ref{rdf_SI} shows the reduced PCF obtained
from the MD models of {{\it a}-Si}, which consist of 400,000
atoms. The data presented here correspond
to the configurational-averaged values of $G(r)$
from three independent configurations. The inset
in Fig.\,\ref{rdf_SI} shows the presence of distinct
radial oscillations at a distance beyond 20 {\AA},
extending at least up to 40 {\AA}.
Similar oscillations have been also observed in
the reduced PCF of 21,952-atom WWW models, but
in a somewhat weaker form.
\begin{figure}[t!]
\centering
\includegraphics[width=0.45\textwidth]{FIG1}
\caption{
The pair-correlation functions of three partially-ordered
models (M1 to M3) of Si atoms, showing radial correlations
up to a length of 6 {\AA}. The results for {{\it a}-Si} (WWW models)
are shown for comparison with that for the M3 model. The
size of the models corresponds to 21,952 atoms and the PCF
data were averaged over three independent configurations
for each model.
}
\label{rdf_M}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=0.45\textwidth]{FIG2}
\caption{
The reduced pair-correlation function, $G(r)$, of {{\it a}-Si},
obtained from a configurational averaging of three large
MD models of size 400,000 atoms. The inset shows
the presence of radial oscillations up to 40 {\AA},
which are known as the extended-range oscillations
in {{\it a}-Si}.
}
\label{rdf_SI}
\end{figure}
This is apparent in Fig.\,\ref{400MD}, where we
have plotted the configurationally averaged reduced
PCFs for the 400,000-atom MD models and 21,952-atom
WWW models.  For comparison, the radial distances
($r$) in Fig.\,\ref{400MD} are scaled by the
corresponding position of the first peak ($r_0$)
by introducing a scaled variable $R = r/r_0$.  The
inset in Fig.\,\ref{400MD} clearly shows the
presence of considerable oscillations in larger
400,000-atom MD models compared to their WWW
counterpart in the region of $R$ from 6 to 14, which
translates into a distance of 14 {\AA} to 33 {\AA}
for $r_0 \approx$ 2.37 {\AA}. The observed differences
can be partly attributed to the size and statistics
and in part to the nature of simulations. In
general, MD models are considered to be more
representative of annealed samples of {{\it a}-Si}, which
are slighly more ordered than their as-deposited
counterpart.

Table \ref{tab1} presents some
characteristic structural properties of the MD and
WWW models. Since the presence of too many structural defects
can affect the local density of the networks,
and the radial correlations between atoms, it
is necessary for the models to exhibit properties
that are compliant with experimental observations.
The presence of only a few dangling bonds
(up to 1.3\%) and floating bonds (up to 1.2\%),
as well as a small value of the root-mean-square width,
$\Delta \theta$, about 9--10{$^\circ$}, of the
bond-angle distribution, confirms that the
structural properties of these models are indeed
consistent with actual samples of {{\it a}-Si}.

To further characterize the models, one often
computes the electronic density of states (EDOS).
The EDOS in {{\it a}-Si} is found to be very
sensitive to the presence of coordination
defects, especially three-fold-coordinated Si
atoms or dangling bonds.
The presence of an electronic gap largely
depends on these defects, and the size of
the gap is known to be related to the
density of such defects and the degree of
disorder in bond-length and bond-angle
distributions. We have therefore calculated
the EDOS of 21,952-atom WWW models and
400,000-atom MD models.  Since the diagonalization
of the Hamiltonian matrix ($H$) of such
large {{\it a}-Si} models is highly nontrivial, we
had to resort to: a) the tight-binding approximation
of the Hamiltonian; and b) employ the recursion
method of Haydock, Heine, and Kelly (HHK)~\cite{Haydock:1972,Haydock:1980} to
obtain the EDOS. In the recursion approach of
HHK, one calculates the projected density of states
$n_{\alpha}(E)$, associated with a basis function
$|\alpha\rangle$ (involving a site and an orbital),
by writing
\begin{equation}
n_{\alpha}(E)= \sum_k |\langle \alpha|\psi_k\rangle|^2 \delta(E-E_k).
\end{equation}
Here, $E_k$ and $\psi_k$ are the
energy eigenvalues and eigenvectors
of $H$, respectively. Using a
representation of the $\delta$-function
and writing $z = E + \imath\epsilon$, where
$\epsilon \to 0^{+}$,  it can be shown that the projected
EDOS can be expressed in terms of the
singular part of the diagonal element
of the resolvent of $H$ or the Green's operator
$ \hat{G}(z) = (z\hat{I}-\hat{H})^{-1}$. This
yields~\cite{green}
\begin{equation}
n_{\alpha}(E) = -\frac{1}{\pi} \lim_{\epsilon \to 0^+} \Im \, G_{\alpha\alpha}(E + \imath\epsilon).
\label{LDOS}
\end{equation}
The local EDOS obtained from using Eq.~(\ref{LDOS}) is
averaged over multiple sites to calculate the
total EDOS. For 400,000-atom MD models, the problem is particularly
difficult due to the handling and storage of large matrices and the
computational cost associated with the calculation for
all sites. In practice, a few clusters of several
hundred atoms are found to suffice for configurational
averaging. Using a fast matrix-vector multiplication scheme
and a compressed representation of the
sparse $H$ matrix, one can implement an
order-$N$ algorithm for the calculation of the
local EDOS in the tight-binding approximation.
The results obtained from these calculations
are shown in Fig.\,\ref{EDOS}. The presence
of a clean gap, rather than a pseudo gap, in
the EDOS further establishes
the quality of the models. The approach can
be adapted to calculate the vibrational density
of states in the harmonic approximation,
provided that an efficient scheme to obtain
electronic forces for the construction of the
dynamical matrix (DM) of {{\it a}-Si} is available.
An accurate order-$N$ approach to construct
the DM within the framework of tight-binding
formalism can be found in Ref.~\onlinecite{Biswas:2002}.

\begin{figure}[t!]
\centering
\includegraphics[width=0.45\textwidth]{FIG3}
\caption{
The reduced PCFs of the 400,000-atom MD models
and 21,952-atom WWW models showing the presence
of considerable extended-range oscillations in
larger MD models. For clarity, the radial
distances are scaled by the corresponding first
peak of the PCF, i.e., $R = r/r_0$, where
$r_0$=2.37 {\AA}.
}
\label{400MD}
\end{figure}

\begin{figure}[t!]
\centering
\includegraphics[width=0.45\textwidth]{FIG4}
\caption{
The electronic density of states (EDOS)
of 21,952-atom WWW and 400,000-atom MD
models obtained from the tight-binding
approximation. The presence of an electronic
band gap is clearly visible in the plots.
}
\label{EDOS}
\end{figure}

\begin{table}[t!]
\caption{
Structural properties of three WWW models (W1--W3)
and three MD models (MD1--MD3). The average bond
length ($\langle r \rangle$), average bond angle
($\langle \theta \rangle$), and the root-mean-square
width of bond angles ($\Delta \theta$) are expressed
in {\AA} and degree, respectively. $C_n$ indicates
the number of $n$-fold-coordinated atoms
(in percent). \\
}
\centering
\begin{tabular}{ c| c|c | c | c | c|c  |c|c  }
\hline
 \multicolumn{2}{c|} {Model}& \multicolumn{2}{c|} {Bond angle} &  \multicolumn{4}{c|} {Atomic coordination}& Bond length \\
 \cline{1-8}
Type & Size $(N)$ & $\langle \theta \rangle$ & $\Delta \theta$ & $C_{2}$&$C_{3}$ &$C_{4} $ & $C_{5} $&$\langle r \rangle$  \\
\hline
W1  & 21,952  &109.21  &10.04 & 0.00 & 0.00 & 99.86 & 0.14 & 2.36 \\
W2  & 21,952  &109.23  &9.83  & 0.00 & 0.00 & 99.9  & 0.1  & 2.36\\
W3  & 21,952  &109.22  &9.87  & 0.00 & 0.00 & 99.88 & 0.12 & 2.36\\
MD1 & 400,000 &109.23  &9.26  & 0.02 & 1.28 & 97.59 & 1.11 & 2.38 \\
MD2 & 400,000 &109.23  &9.31  & 0.03 & 1.29 & 97.52 & 1.16 & 2.38 \\
MD3 & 400,000 &109.23  &9.34  & 0.02 & 1.26 & 97.57 & 1.15 & 2.38 \\
\hline
\end{tabular}
\label{tab1}
\end{table}

\subsection{Origin of extended-range oscillations in {{\it a}-Si}}

The first step toward understanding the ERO in {{\it a}-Si}
follows from an analysis of the reduced PCF of
disordered crystalline silicon ({{\it dc}-Si}) structures.
The inclusion of positional disorder washes out the
sharp ${\delta}$-functions in the PCF of diamond
{{\it c}-Si} and leads to a series of broadened peaks for
the resulting {{\it dc}-Si} structures. A comparison of the
reduced PCF of {{\it a}-Si} with those from {{\it dc}-Si},
for $\sigma$ = 1.0 {\AA} and 1.2 {\AA}, in Fig.\,\ref{rdf_CSI}
reveals that {{\it a}-Si} exhibits small but noticeable
oscillations at large distances
of up to at least 30 {\AA}. Despite the fine structure of
$G(r)$ in {{\it dc}-Si}, it is apparent that the positions
of the peaks in {{\it a}-Si} approximately coincide with
those in {{\it dc}-Si}. This observation leads to the
possibility that the ERO in {{\it a}-Si}
could originate from the presence of weak radial-shell
structures on the nanometer length scale, as in the
case of {{\it dc}-Si}. This point is examined at length in
the following paragraphs.

\begin{figure}[t!]
\centering
\includegraphics[width=0.5\textwidth]{FIG5}
\caption{
The presence of radial oscillations in MD models
of {{\it a}-Si} (blue) of size 400,000 atoms and two {{\it dc}-Si}
structures of size 405,224 atoms from 15 {\AA}
to 30 {\AA}. The positions of the radial peaks of
{{\it a}-Si} approximately correspond to those of {{\it dc}-Si},
indicating the possible presence of weak extended-range
ordering in {{\it a}-Si} beyond 15 {\AA}.
}
\label{rdf_CSI}
\end{figure}

Assuming that radial-shell structures exist in the
partially-ordered environment of {{\it a}-Si} at large
distances, one may express the total PCF, $g(r)$,
as a linear combination of the same for each
coordination shell, $g_n(r)$. Thus, $g(r) = \sum_n g_n(r)$,
where $g_n(r)={\langle g(r=|\mathbf{r_n-R_i}|)\rangle}_i$.
Here, $r$ is the distance between a central atom at
$\mathbf R_i$ and its neighbors in the $n$th coordination
shell at ${\mathbf r_n}$, and the symbol
${\langle \, \rangle}_i$ stands for the average over all
atoms and independent configurations.
Since, for an arbitrary (highly) disordered network, distant radial
shells may not exist or be well defined -- depending on the
degree of radial disorder -- it is more appropriate to
define the $n$th coordination or topological shell as
one that consists of $n$th near neighbors of the central
atom at $\mathbf{R_i}$.
\begin{figure}[ht]
\centering
\includegraphics[width=0.3\textwidth]{FIG6}
\caption{
A schematic representation showing the first two
coordination shells of a central atom (blue) in
a two-dimensional disordered network. The atoms
in the first shell (green) and the second shell
(yellow) can be reached from the central
atom in one step and two steps, respectively.
}
\label{2D}
\end{figure}
This is schematically illustrated in Fig.\,\ref{2D} by
showing the first-shell neighbors (green) and the
second-shell neighbors (yellow) of the central atom (blue).
The key point here is that the $n$th
neighbors of a central atom are those that can be reached
(from the center) by a minimum of $n$ distinct and
irreversible steps, irrespective of the presence of
well-defined radial shells or not.  Thus, the
coordination shells defined above depend on the
topology or connectivity of the atomic network, and
the three-dimensional shape of the shells may not
be necessarily spherical.  We shall see later that this
can lead to a highly asymmetrical radial distribution of atoms within
the coordination shells of partially-disordered networks.
Figure \ref{shell} shows the shell PCFs, $g_n(r)$,
obtained for the first six coordination shells,
along with $g(r)$ for a 21,952-atom WWW model of {{\it a}-Si}.
It is apparent that the shell PCFs, for $n$ = 1 to $n$ = 6,
can be represented by a bell-shaped curve in
{{\it a}-Si}, with the exception of $g_3(r)$ for
which a bi-modal distribution is observed.
The latter is consistent with the
earlier study by Uhlherr and Elliott~\cite{Uhlherr:1994}.
The bi-modal shape of $g_3(r)$ originates from the
distribution of the end-to-end radial distances of
a set of four neighboring atoms or quartets associated
with dihedral angles in {{\it a}-Si}.

\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{FIG7}
\caption{
The shell pair-correlation function, $g_n(r)$,
for the $n$th coordination shell of a 21,952-atom
WWW model of {{\it a}-Si}.  The total $g(r)$ (dashed
black), which is given by the sum of all
shell PCFs, is also shown in the plot.
}
\label{shell}
\end{figure}

Having expressed the total PCF in terms of $g_n(r)$,
we now examine the oscillations in the PCF by
studying individual $g_n(r)$s, which reflect the
characteristic properties of the radial distributions
of atoms in $n$th shells.  In particular, the width
of $g_n(r)$ is indicative of the strength of the
radial (dis)order in the $n$th shell. A small value of
the width corresponds to a highly ordered state of
atoms within the shell as far as radial ordering is
concerned, and vice versa. This assertion can be
verified by computing $g_n(r)$ for a number of
partially-ordered networks of silicon.  Figure
\ref{g13} shows the results for the 13th coordination
shell, $g_{13}(r)$ as a representative example,
obtained from 21,952-atom models,
of {{\it a}-Si}, {{\it dc}-Si}, and M2.  As stated earlier in Sect.\,II,
the latter model (M2) is characterized by the presence
of a well-defined first-coordination shell,
whereas the {{\it dc}-Si} structures are produced by using
a value of $\sigma$ in the range from 0.3 {\AA}
to 1.0 {\AA}.
It is apparent that a small value of $\sigma$ (for
example, $\sigma$ = 0.3 {\AA}) produces well-defined
multiple peaks in $g_{13}(r)$ for {{\it dc}-Si} models. However,
as the value of $\sigma$ increases and goes beyond 0.6 {\AA},
the peaks in $g_{13}(r)$ coalesce to form a unimodal
distribution. This is unsurprising due to the presence of
strong residual crystalline order in the {{\it dc}-Si} networks
for $\sigma \le$ 0.6 {\AA}. By contrast, the width of
$g_{13}(r)$ for {{\it a}-Si} is found to be considerably smaller
than its M2 counterpart, which shows a more radially disordered
distribution of atoms within the same shell in M2. This
observation is found to be true not only for $g_{13}(r)$
but also for all $g_n(r)$s.
The high asymmetry of $g_{13}(r)$ for the M2 and {{\it dc}-Si}
models can be readily attributed to the connectivity of
the atoms in these models. Since the position of an atom
in a given coordination shell is determined by the number
of steps/hops from the central atom, there exist a few
atoms in the shell that are radially close to the central
atom but are not reachable (from the central atom) via a
small number of steps/hops, due to the low connectivity
of the atoms in the networks for increasing values of
$\sigma$. This is reflected in the left tail of the
distribution (see Fig.\,\ref{g13}), which leads to a
negative value of the skewness for the radial distribution
of atoms in the shell. This can be verified by computing
the Fisher-Pearson (FP) coefficient of skewness~\cite{Fisher},
$s$, for $g_{13}(r)$, for different $\sigma$ values.
In general, the FP coefficient of skewness is given
by the standardized third central moment of a
distribution, and a negative value of the coefficient
signifies a skewed distribution toward the left,
and vice versa.
The variation of $s$ with $\sigma$ for a number of {{\it dc}-Si}
models is shown in Fig.\,\ref{skew}, along with the
corresponding value of the coefficient for the M2
model for comparison.

\begin{figure}[t!]
\centering
\includegraphics[width=0.5\textwidth]{FIG8}
\caption{
The shell pair-correlation functions, $g_{13}(r)$,
obtained from 21,952-atom WWW models of {{\it a}-Si} (red),
{{\it dc}-Si} (blue/purple/cyan), and partially-ordered
configurations M2 (green) of Si atoms. The disordered
crystalline structures were generated from the
diamond {{\it c}-Si} structure, using $\sigma$ = 0.3,
0.8, and 1.0 {\AA}.
}
\label{g13}
\end{figure}

\begin{figure}[ht!]
\centering
\includegraphics[width=0.5\textwidth]{FIG9}
\caption{
The variation of the Fisher-Pearson coefficient
of skewness, $s$, for the distribution $g_{13}(r; \sigma)$
with $\sigma$ for a number of {{\it dc}-Si} models (blue).
The coefficients for the M2 model (green) and
{{\it a}-Si} (red), for $\sigma$ = 0, are shown in the
plot for comparison.
}
\label{skew}
\end{figure}

\begin{figure}[t!]
\centering
\includegraphics[width=0.5\textwidth]{FIG10}
\caption{
The full width at half maximum (FWHM) for a set of
partially-ordered configurations (M1 to M3), {{\it a}-Si},
and four {{\it dc}-Si} structures of size 21,952 atoms.
The {{\it dc}-Si} structure (with $\sigma$ = 0.2 {\AA})
is the most ordered configuration, whereas M1 is
the least ordered configuration, by construction.
}
\label{width}
\end{figure}

Figure \ref{width} shows the full width at half maximum
(FWHM) of the shell PCFs, $g_n(r)$, for different
shells, from $n$=1 to $n$=20, for a class of partially-ordered
models (M1 to M3), {{\it a}-Si}, and {{\it dc}-Si} models of size 21,952
atoms. Since {{\it dc}-Si} models tend to exhibit the presence of
multiple peaks in $g_n(r)$ for $\sigma \le $ 0.6 {\AA},
the FWHM for the {{\it dc}-Si} models (with multiple peaks) in
Fig.\,\ref{width} is calculated by fitting each individual
peak with a Gaussian distribution and averaging over the
resulting FWHM values for all major peaks in the distribution.
The FWHM values (in Fig.\,\ref{width}) suggest that
the M1 models are highly disordered, whereas the {{\it dc}-Si}
structures with $\sigma$ = 0.2 {\AA} are the least
disordered configurations. This observation is
indeed true by construction. For $\sigma$ = 0.2--0.5
{\AA}, a significant radial ordering exists in the
{{\it dc}-Si} structures that leads to a small value of
the width in Fig.\,\ref{width}. The rest of the
models, from M2 and M3 to {{\it a}-Si}, exhibit an
increasingly more ordered state of radially distributed
atoms in the shells. It is apparent that as more
radial ordering is incorporated in a model
(for example, M2 and M3), the corresponding FWHM
value of $g_n(r)$ begins to decrease for a given
shell. Conversely, the inclusion of (additional) structural
disorder increases the corresponding FWHM value
of $g_n(r)$ in a model. This can be seen from
Fig.\,\ref{d-width1}, where the addition of positional
disorder, via random displacements of atoms in M1, M3,
and {{\it a}-Si}, resulted in an increase of
the FWHM values of $g_n(r)$. This observation also
applies to the total PCF of {{\it a}-Si}. Figure \ref{d-width2}
shows that the amplitude of the radial oscillations
reduces in the region of 20--40 {\AA} with the
addition of positional disorder in {{\it a}-Si}.
It may be noted that the FWHM values for the M1 models,
which are highly disordered by construction, are
practically unaffected in Fig.\,\ref{d-width1}
in the presence of additional disorder with $\sigma$
values of the order of 0.3 {\AA}.
Thus, the width (or the average width for a multimodal
case) of $g_n(r)$ can be taken as a measure of the
radial order/disorder in partially-ordered networks,
including {{\it a}-Si} and {{\it dc}-Si} structures.

\subsection{Shannon information as a measure of
extended-range ordering}

The assertion that the width of the shell pair-correlation
function, $g_n(r)$, can provide a measure of the
disorder in the radial distribution
of atoms in the $n$-th coordination shell of a disordered
network is not particularly surprising and it directly
follows from the Shannon measure of information
(SMI)~\cite{Shannon:1948}. By normalizing the
shell PCF, $g_n(r)$, one can readily construct a
discrete probability measure, $p_n^i$, to define
the SMI
\begin{equation}
S[p_n^i] = -k\, \sum_i p_n^i \, \ln p_n^i.
\label{eq2}
\end{equation}
In Eq.(\ref{eq2}), the value of $0\ln(0)$ is
defined to be 0, $k$ is a constant, and $p_n^i$
is given by
\[
p_n^i = \frac{g_n(r_i)}{\sum_i g_n(r_i)}.
\]
The SMI can be understood as providing a measure
of the degree of uncertainty or the lack of
radial ordering in the distribution of atoms in
the coordination shells. The multiplicative
constant $k$ in Eq.\,(\ref{eq2}) can be taken
as unity without any loss of generality.
The results for the SMI obtained from M1/M2/M3/{{\it a}-Si}
models are shown in Fig.\,\ref{SMI} for the first
twenty coordination shells. The corresponding results
for diamond {{\it c}-Si} are also shown in the plot
for comparison. As one may expect, the SMI values for
different shells behave in a similar manner as that
of the FWHM (of the shell PCFs) with respect to
the peak position in Fig.\,\ref{width}. Once again, the largest
values of the SMI correspond to the highly
disordered M1 models, whereas {{\it a}-Si} exhibits the
smallest values of the SMI for each shell among M1, M2,
M3 and {{\it a}-Si}. It is noteworthy that, unlike the case of
disordered and amorphous Si networks,  the SMI values
associated with the coordination shells in the
diamond {{\it c}-Si} structure, which is perfectly ordered, increase
considerably with the increasing shell number in a
global sense. This observation can be attributed to
the presence of multiple peaks in
the higher-order coordination shells. Since the (shell)
PCFs for a crystalline structure consist of a series
of $\delta$-functions, the presence of an increasing
number of peaks in the distant
shells leads to more uncertainty in the radial distribution
of the atoms in these shells.  This
is reflected in the larger value of the SMI for
the distant shells. Thus, the SMI can be loosely
interpreted as a global measure of ordering/disordering in
the distribution, which is most appropriate for
describing the degree of order/disorder associated
with unimodal distributions.
However, for multimodal distributions, such
as the {{\it dc}-Si} structures with $\sigma \le$ 0.5 {\AA},
one requires a suitable local measure of information,
for example, the Fisher information~\cite{Fisher:1959},
in order to quantify the degree of disorder or uncertainty
associated with the radial distribution of the atoms in the
coordination shells.
These issues will be addressed elsewhere from
an information-theoretic point of view in a
future communication.

\begin{figure}[t!]
\centering
\includegraphics[width=0.5\textwidth]{FIG11}
\caption{
The effect of the addition of positional disorder
on the FWHM of the shell PCFs in M1, M3, and {{\it a}-Si}.
The FWHM values of the unperturbed models are
also shown for comparison.
}
\label{d-width1}
\end{figure}

\begin{figure}[t!]
\centering
\includegraphics[width=0.5\textwidth]{FIG12}
\caption{
The effect of the addition of positional disorder, with
$\sigma$ = 0.5 {\AA}, on the radial oscillations in a
400,000-atom MD model of {{\it a}-Si} between 20 {\AA} and
40 {\AA}.  The results for the corresponding pristine
{{\it a}-Si} model (green) are also shown for comparison.
}

\label{d-width2}
\end {figure}

\begin{figure}[t!]
\centering
\includegraphics[width=0.5\textwidth]{FIG13}
\caption{
The Shannon measure of information (SMI),
associated with a discrete probability measure
$p_n$, obtained from the shell pair-correlation
functions, $g_n(r)$, for M1 to M3, {{\it a}-Si}, and
diamond {{\it c}-Si}. For disordered and amorphous
Si networks, the results were averaged over
three independent configurations for each shell.
}
\label{SMI}
\end{figure}

The origin of the extended-range oscillations in {{\it a}-Si}
can now be interpreted in light of the results from
Figs.\,\ref{shell}--\ref{SMI}. Since the full PCF
can be expressed in terms of its partial components,
any structural aspects of $g(r)$, such as the
extended-range oscillations, can also be represented
by a suitable set of $g_n(r)$, associated with the
length scale of the oscillations. Figures
\ref{width} and \ref{SMI} essentially suggest that,
as the degree of radial ordering in the full PCF
increases from M1 to M3, the corresponding width
and the Shannon information associated with $g_n(r)$
steadily decrease.
Thus, the inclusion of radial information of up to a
distance of 4 {\AA} in M2 and about 6 {\AA} in M3
suffices to result in a reduction of the width of
$g_n(r)$ associated with the distant coordination
shells. Since {{\it a}-Si} is characterized by the presence
of strong radial ordering at least up to a length
of 20 {\AA} in the full PCF, it is unsurprising
that a small value of the width of $g_n(r)$ of
{{\it a}-Si} is reflective of the radial ordering in
the distant shells on the length scale of 20--40 {\AA}.
By contrast, the {{\it dc}-Si} models with $\sigma$ = 0.2--0.5
{\AA} show significant radial ordering as far as
the widths of various $g_n(r)$s are concerned.
Thus, the ERO in {{\it a}-Si} can be understood as the
resultant density fluctuations, originating from
highly ordered radial distributions of atoms in
the first few coordination/radial shells, which propagate
and decay radially as the (density) fluctuations
travel through the distant shells.  A comparison
of the results from the M2, M3, and {{\it a}-Si} models
in Fig.\,\ref{width} appears to suggest that
the characteristic local radial ordering of up to
6 {\AA} forces the atoms in distant shells to
organize in such a way that small radial oscillations
are built up on the length scale of up to 40 {\AA},
when the model is sufficiently large.
The presence of these small but distinct radial
oscillations in the full PCF is indicative of
the existence of weak extended-range radial
ordering in {{\it a}-Si} up to a length of 40 {\AA},
as far as the size of the {{\it a}-Si} models studied
in this work are concerned.

\begin{figure}[t!]
\centering
\includegraphics[width=0.5\textwidth]{FIG14}
\caption{
The autocorrelation coefficient, $\gamma(r_k)$,
of a set of $G(r)$ values from $r$ = 15 {\AA} to
$r$ = 45 {\AA}, constructed from 400,000-atom
MD models of {{\it a}-Si}, showing the presence of
radial correlations up to 45 {\AA}. The root-mean-square
fluctuations are shown as error bars (blue vertical
lines). For visual clarity, the results for the radial
region from 35 {\AA} to 45 {\AA} are shown in
the inset.
}
\label{coreal}
\end {figure}

\subsection{
Decay of radial correlations, autocorrelation
coefficient, and comparison with experimental
diffraction data
}

The presence of radial atomic correlations
beyond 20 {\AA} can be further evidenced by
computing the autocorrelation coefficient(s)
of $G(r)$. Assuming that $M$ observations,
$y_1$, $y_2$, \ldots, $y_M$, form a time series,
where $y_i = G(r_i)$, the autocovariance
coefficient~\cite{cov}, $c_k$, between the
observations that are $k$-steps apart, is
given by:
\begin{equation}
c_k= \frac{1}{M} \sum_{i = 1}^{M-k} (y_i - \bar{y}) (y_{i+k} - \bar{y}),
\: k = 1,...,n, n < M.
\label{eq3}
\end{equation}
The autocorrelation coefficient at lag $k$ is then expressed as
$\gamma_k = c_k/c_0$, where $c_0$ is the variance and
${\bar y}$ is the mean value of the set $\{y_i\}$.
Figure \ref{coreal} shows a plot of $\gamma_k$
versus $r_k$.
Here, the set $\{y_i\}$ is constructed by choosing a
segment of $ G(r)$ from $r_1$ = 15 {\AA} to $r_n$ = 45 {\AA}
and expressing $k$ in terms of $r_k = r_1 + k\Delta r$, where
$\Delta r$ is the distance between two consecutive observations
of $G(r)$.
It is apparent from the plot that, given the set of $G(r)$ values
from 15 {\AA} to 45 {\AA}, $\Delta r$ = 0.05 {\AA},
$n$ = 600, and $M$ = 900, the radial correlations decay in an
oscillatory manner and become almost negligible after
35 {\AA}. The root-mean-square (RMS) fluctuations of $\gamma_k$,
obtained from the configurational averaging of the results
from three independent MD models of size 400,000 atoms, are
also shown in Fig.\,\ref{coreal}. Since the RMS values of the
fluctuations are almost of the order of $\gamma_k $ for
$ r \ge $ 40 {\AA}, the radial correlations in this region may
not be significant, even though the presence of small
residual correlations can be seen in this region.

We now provide a direct comparison of our results
with those from diffraction measurements by computing the
decay length and the (spatial) period of the
ERO at radial distances beyond 10 {\AA}.
High-energy x-ray diffraction measurements
on {{\it a}-Si} samples, by Roorda et al.~\cite{Roorda:2012},
suggest that the period of oscillations ranges
from 2.77 {\AA} to 3.03 {\AA} and that the
decay length in annealed samples of {{\it a}-Si}
is about 4.23 {\AA}.
Figure \ref{decay} shows the decay of the amplitudes of
radial oscillations in $G(r)$ for 400,000-atom MD models
of {{\it a}-Si}, which can be roughly considered as the
simulated counterpart of annealed samples of {{\it a}-Si}
in experiments. Here, the amplitudes and positions
of the peaks are obtained from Fig.\,\ref{rdf_SI}.
The corresponding decay for {{\it dc}-Si}
networks for $\sigma$ = 0.7 {\AA} and 0.8 {\AA} are
also included in the plot for comparison. For visual
clarity, the first three peaks of {{\it a}-Si} are omitted
from the plot, by choosing an appropriate range for
the $y$ axis. The values of the period and the decay
length obtained from our calculations compare very
well with the results from experiments.
The average period of oscillations from 400,000-atom
MD models, in Fig.\,\ref{decay} (and Fig.\,\ref{rdf_SI}),
is found to be 3.2$\pm$0.065 {\AA}, which is very
close to the experimental value of 3.03 {\AA}, and
the corresponding decay length turns out to be about
4.81$\pm$0.012 {\AA}. The latter is somewhat higher
than the experimental value of 4.23 {\AA}, obtained
from the Fourier transform of experimental diffraction
data by~\citet{Roorda:2012}. It is evident from
Fig.\,\ref{decay} that the {{\it dc}-Si} models exhibit
a rather slow decay, even for considerably large
values of $\sigma$ from 0.7 {\AA} to 0.8 {\AA}.

\begin{figure}[t!]
\centering
\includegraphics[width=0.5\textwidth]{FIG15}
\caption{
The decay of the radial peak heights in $G(r)$ for {{\it a}-Si}
(blue) and {{\it dc}-Si} (red and green) with
peak distances. The exponential fit of the peak
positions (dashed blue line) corresponds to the
data for {{\it a}-Si} models of size 400,000 atoms.
The decay length for {{\it a}-Si} corresponds to a
value of 4.81 {\AA}. The corresponding peak positions
for {{\it dc}-Si} structures are shown for comparison.
}
\label{decay}
\end {figure}

\subsection{Relation between ERO and the first sharp
diffraction peak in {{\it a}-Si}}

In this section, we address the question whether the
presence of extended-range oscillations has any
bearing on the position and intensity of the first sharp diffraction
peak (FSDP) in {{\it a}-Si}.
\begin{figure}[t!]
\centering
\includegraphics[width=0.5\textwidth]{FIG16}
\caption{
The static structure factor, $S(Q)$, of a 21,952-atom WWW
model of {{\it a}-Si} obtained from the Gaussian approximation
(red line) and from direct numerical calculations
(blue circles) using Eq.\,(\ref{eq5}).
}
\label{SQ-approx}
\end{figure}
Since the origin of the FSDP is
strongly related to the presence of medium-range order
(MRO) in glasses, which can extend up to a radial
distance of approximately 20 {\AA}, it is instructive
to examine whether the ERO in {{\it a}-Si} can produce any
observable effect on the intensity of the FSDP near
2.0 {\AA}$^{-1}$. Noting that the oscillations are particularly pronounced
in $G(r)$, it is useful to write $G(r)$ as a linear
combination of Gaussian functions,
\begin{equation}
G(r) = \sum_{i=1}^m a_i e^{-b_i (r - c_i)^2},
\label{eq4}
\end{equation}
in an effort to obtain an analytical expression
for $S(Q)$ in terms of the Gaussian parameters.
The parameters $a_i$, $b_i$, and $ c_i$ determine the
approximate peak/trough height, width, and the (radial)
position of the $i^{\text{th}}$ peak/trough, respectively,
and can be obtained either via a nonlinear fit of
Eq.\,(\ref{eq4}) to experimental/simulated reduced
PCF data, $G(r)$, or by minimizing a suitable cost
function with respect to the set of parameters $(a_i,
b_i, c_i)$. Here, we have taken the second approach
and ensured that $b_i > 0$ for all $i$. The structure
factor can be expressed in terms of the fitted Gaussian
parameters:
\begin{eqnarray}
S(Q) & = & 1 + \int_0^{R_c} rG(r) \,\frac{\sin(Qr)}{Qr} \, dr  \label{eq5} \\
     & = & 1 + \frac{1}{Q} \sum_{i=1}^{m} a_i\sqrt{\frac{\pi}{b_i}}\, \sin(Qc_i)
\, \exp\left[-\frac{Q^2}{4b_i}\right]
\label{eq6}
\end{eqnarray}
\noindent

In writing Eq.\,(\ref{eq6}), we have denoted, for notational
convenience, the set $(a_i, b_i, c_i)$ as the fitted
values of the parameters and assumed that the center of
each Gaussian function, $c_i$, satisfies the condition
$0 \ll c_i \ll R_c$ so that $S(Q)$ can be written as a
sum of Gaussian integrals (and not error functions) with
the integration limit extending from 0 to $\infty$.
This condition is readily satisfied by choosing an
appropriate value of $m$, such that $R_c \gg c_m$,
and noting that the first peak of the PCF in {{\it a}-Si}
rapidly decays to zero for $r \le 2.0$ {\AA}.
In practical calculations, a value of $R_c$ of the order
of 20 {\AA} is found to be sufficient for accurate
determination of $S(Q)$ using Eq.\,(\ref{eq5})
[see \citet{Dahal:2021} and Fig.\,\ref{conv} here].
The structure factor obtained from
Eq.\,(\ref{eq6}) for a 21,952-atom WWW model of {{\it a}-Si} is
plotted in Fig.\,\,\ref{SQ-approx}, along with the results
from direct numerical calculations from Eq.\,(\ref{eq5})
for comparison. For clarity, the wavevector region from
15 {\AA}$^{-1}$ to 30 {\AA}$^{-1}$ is shown separately
as an inset in Fig.\,\ref{SQ-approx}.

\begin{figure}[t!]
\centering
\includegraphics[width=0.5\textwidth]{FIG17}
\caption{
The convergence of the intensity of the FSDP (at $Q_0$)
and the principal peak (at $Q_1$), obtained from
Eq.\,(\ref{eq5}), with respect to the number ($i$)
of Gaussian peaks/troughs for a 21,952-atom WWW model.
The radial length associated with the Gaussian
peaks/troughs is indicated in {\AA} on the secondary
$x$ axis (top).
}
\label{conv}
\end {figure}

The variation of the intensity of the FSDP
and the principal peak (i.e., the peak at
3.6 {\AA}$^{-1}$) can be studied, by using
Eq.\,(\ref{eq6}), with respect to the number of Gaussian basis
functions $m$ for a given $R_c$.  Writing
$\Delta S(Q,i)=S(Q,m)-S(Q,i)$, where $m$=70
for $R_c$ = 30 {\AA}, Fig.\,\ref{conv} shows
the convergence of $\Delta S$ at $Q_0$ = 1.94
{\AA}$^{-1}$ and $Q_1$ = 3.6 {\AA}$^{-1}$ for
an increasing number ($i$) of peaks/troughs.
Here, $Q_0$ and $Q_1$ correspond to the position
of the FSDP and the principal peak, respectively.
It is apparent that both $S(Q_0,i)$ and $S(Q_1,i)$
converge to the respective limiting value, $S(Q, m)$,
very rapidly as $i$ approaches to 30, which corresponds
to a radial length of about 18 {\AA}, as indicated
in Fig.\,\ref{conv}. The length is indicated at
the top of the plot as a secondary $x$ axis,
which reflects the non-uniform distribution of
Gaussian peaks/troughs in the radial-region of
0--30 {\AA}. Figure \ref{conv} suggests that
radial correlations from the region beyond 20 {\AA}
do not really play any significant role. This
observation can be stated more precisely. The
magnitude of the contribution to $S(Q)$ obtained
by including an additional peak/trough beyond
$m$ in Eq.\,(\ref{eq6}) can be written as:
\begin{eqnarray}
|\delta S(Q,m)| &=& |S(Q,m+1)-S(Q,m)| \notag \\
&=& \left| \frac{1}{Q} \sqrt{\frac{\pi a_{m+1}^2}{b_{m+1}}}\,
\sin(Q c_{m+1}) \, \exp\left[-\frac{Q^2}{4b_{m+1}}\right]\right | \notag \\
&\le&
\frac{1}{Q} \sqrt{\frac{\pi a_{m+1}^2}{b_{m+1}}} \, \exp\left[-\frac{Q^2}{4b_{m+1}}\right].
\label{eq7}
\end{eqnarray}
Substituting $Q$ = $Q_0$ = 2 {\AA}$^{-1}$ in Eq.\,(\ref{eq7})
for the FSDP in {{\it a}-Si}, one obtains:
\begin{equation}
|\delta S(Q_0, m)| < \frac{a_{m+1}}{\sqrt{b_{m+1}}} \,
\exp\left[-\frac{1}{b_{m+1}}\right]
\label{eq8}
\end{equation}
The asymptotic behavior of $|\delta S(Q_0,m)|$ with respect to $m$
follows from Eq.\,(\ref{eq8}). Since $G(r) \rightarrow$ 0 as $r
\rightarrow R_c$ for very large models, the parameter $a_m$, which
determines the height of the Gaussian peak, decreases with an
increasing value of $m$, and $|\delta S(Q_0,m)|$ becomes
increasingly smaller as $m$ becomes a large number.  In practice, however,
$|\delta S(Q_0,m)|$ fluctuates between 0 and a small value $\epsilon$ due to
the presence of numerical noise at large radial distances,
which can be reduced by averaging $S(Q)$ [in Eq.\,(\ref{eq6})]
over many independent sets of fitted Gaussian parameters.
Further, a value of $R_c$ of about 30 {\AA} is found to be
sufficient for the calculation of $S(Q)$ from
Eqs.\,(\ref{eq5}) and (\ref{eq6}). The results from our calculations
suggest that the average value~\cite{eps} of $\epsilon$ is
typically of the order of 0.025 for radial distances between
20 {\AA} and 30 {\AA}. This roughly translates into an error
of 1.7\%, assuming $S(Q_0)$=1.5 for as-deposited samples from
experiments~\cite{Xie:2013}.  Thus, aside from small fluctuations
of $S(Q_0)$ owing to numerical noise, the extended-range
oscillations in the radial region of 20--30 {\AA} do not
seem to play any observable role in determining the
intensity of the FSDP in {{\it a}-Si}.
A similar conclusion was reached in a recent
study~\cite{Dahal:2021}, where an alternative
argument based on the analysis of the behavior
of $rG(r)$ and the sampling of $\sin(Qr)/Qr$
within the radial region from 0 to $R_c$ in
Eq.\,(\ref{eq5}) was provided by the authors of
the study to support this conclusion.

\section{Conclusions}
The present study addresses the origin of the
extended-range oscillations in {{\it a}-Si} from a
real-space point of view. By analyzing a class
of large partially-ordered networks of Si atoms
with radial ordering up to a distance of 6
{\AA} in the PCF, it has been shown that the
inclusion of short-range ordering in the first
two coordination shells of the disordered
networks can lead to an increased ordering of
the atomic radial distribution in distant
coordination shells. A comparison of these results with those obtained
from large {{\it a}-Si} and disordered crystalline
configurations reveals that the shell pair-correlation
functions for the coordination shells of {{\it a}-Si} at radial
distances of 20--30 {\AA} are considerably
ordered and that this
radial ordering manifests in the form of weak
oscillations in the total PCF of {{\it a}-Si}, which
can be expressed as a sum of the partial radial
distributions from each coordination shell.
By using the full width at half maximum of the peak(s)
of the partial PCFs and the Shannon information
as a measure of the degree of order/disorder, one
arrives at the conclusion that local atomic
correlations can considerably affect the distribution
of atoms in {{\it a}-Si} up to a distance of 40 {\AA}.

An analysis of the amplitude of radial
oscillations in the reduced PCF of 400,000-atom MD
models of {{\it a}-Si} shows that the envelope function of
the reduced PCF decays almost exponentially and
the resulting decay length (of 4.81 {\AA}) is
found to be close to the experimental value
(of 4.23 {\AA}), estimated from the Fourier
transform of the diffraction data obtained
for annealed samples of {{\it a}-Si}.  Likewise, the period of the
extended-range oscillations (for MD models) is
found to be about 3.2 {\AA}, which compares well
with the corresponding experimental value of 3.03
{\AA} for annealed samples.
The study also shows that the structure factor
of {{\it a}-Si} can be expressed as a linear combination
of a series of Gaussian functions, whose
amplitude is modulated by a $sinc$ function.
A convergence study of the intensity of the FSDP,
using the structure factor obtained from the
Gaussian approximation, with respect to the number
of peaks in real space shows that the structure
of the FSDP is primarily determined by the radial
correlations originating from a distance of up
to 20 {\AA} in {{\it a}-Si} networks, which leads to
the conclusion that the ERO has no discernible
effects on the FSDP in {{\it a}-Si}.

\section{Conflicts of interest}
There are no conflicts of interest to declare.

\section{Acknowledgments}
The work was partially supported by the U.S. National
Science Foundation (NSF) under Grant No.\,DMR 1833035.
One of us (P.B.) thanks Dr.\, Raymond Atta-Fynn for
discussions.

\section{References}
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