# Diversified properties of carbon substitutions in silicene

Hai-Duong Phama, Shih-Yang Lina, Godfrey Gumbsb, Nguyen Duy Khanhc, Ming-Fa
Lina,∗ $\\\
\small^{a}DepartmentofPhysics,NationalChengKungUniversity,Tainan701,Taiwan\\\
\small^{b}DepartmentofPhysicsandAstronomy,HunterCollegeoftheCityUniversityofNewYork,\\\
\small 695ParkAvenue,NewYork,NewYork10065,USA\\\
\small^{c}AdvanceInstituteofMaterialsScience,TonDucThangUniversity,19NguyenHuuThoStreet,\\\
\small District7,HoChiMinhCity,Vietnam\\\ $

###### Abstract

The theoretical framework, which is built from the first-principles results,
is successfully developed for investigating emergent two-dimensional (2D)
materials, as it is clearly illustrated by carbon substitution in silicene.
Computer coding with the aid of VASP in conjunction with data analysis from
the multi-orbital hybridizations [spin configurationsf ] are thoroughly
identified from the optimal honeycomb lattices, the atom-dominated energy
spectra, and the spatial charge density distributions. The atom and orbital-
decomposed van Hove singularities [the net magnetic moments], being very
sensitive to the concentration and arrangements of guest atoms. All the binary
2D silicon-carbon compounds belong to the finite- or zero-gap semiconductors,
corresponding to the thoroughly/strongly/slightly modified Dirac-cone
structures near the Fermi level. Additionally, there are frequent $\pi$ and
$\sigma$ band crossings, but less anti-crossing behaviors. Apparently, our
results indicate the well-defined $\pi$ and $\sigma$ bondings.

Keywords: buckled structure, silicene, substitution, geometric structure,
electronic properties.

* Corresponding author. Tel: +886-6-2757575-65272
E-mail address<EMAIL_ADDRESS>(M.F. Lin)

1\. Introduction

Chemical substitutions on layered materials are capable of band structure
tailoring which could lead to significant modifications of the properties of
pristine lattices through very strong host-guest multi-orbital hybridizations.
With the use of modern experimental growth techniques, ternary and binary
compounds, which are characterized by BxCyNz, have been successfully
synthesized for three-dimensional (3D) bulk systems [1, 2], two-dimensional
(2D) layers [3], one-dimensional (1D) cylindrical nanotubes [4], 1D
nanoribbons [5], and zero-dimensional (0D) quantum dots [6]. Their geometric
structures vary from three to zero dimensions, as observed in carbon-related
systems [7, 8]. This clearly indicates that each atom possesses at least three
half-filled orbitals. Similar syntheses have been performed for the high-
potential Si-C compounds [9]. In general, it would be routine to produce the
above-mentioned compounds, whereas the opposite might be true for specific
components. For example, using high-performance experimental techniques, it
may be difficult to manipulate the ratio between the [B, C, N]/[Si, C] atoms.
Such non-monoelement condensed-matter systems have been predicted or found to
exhibit the observable energy gaps, or belong to specific semiconductors. The
main reason lies in the distinct ionization energies of their components,
being consistent with the tight-binding mode for non-vanishing diagonal
Hamiltonian matrix elements [the sublattice-dependent site energies]. Gap
engineering could be achieved by transforming the strength relations in the
$\pi$, $\sigma$ and sp3 bondings the [pz-, (s, px, py)- and (s, px, py,
pz)-orbital hybridizations].

Chemisorption and substitution of silicon atoms on graphene are two very
interesting procedures, since first-principles predictions are available for
understanding the important differences between these two types of chemical
modification [10, 11], and providing additional information regarding another
approach for the formation of silicon-carbide compounds [12]. Based on the
viewpoint of a carbon-created honeycomb lattice, silicon atoms are regarded as
adatoms and guest ones, respectively, in these two cases. The former presents
optimal positions at the bridge sides above the graphene surface [13], while
the latter shows the [Si, C]-co-dominated A and B sublattices with deformed
hexagons [14]. Both silicon and carbon atoms have rather active dangling
bonds, leadings to significant multi-orbital hybridizations in Si-C [Si-Si and
C-C] bonds. Most important, the planar bondings between Si and C atoms are
expected to be stronger and complicated, compared with the perpendicular
configuration. That is to say, the $\pi$ and $\sigma$ [$\pi$] bondings of
graphene will be significantly modified by Si-substitutions [15]. This is
directly reflected in the diversified properties, the spatial charge
distributions, atom-dominated energy bands, and atom- as well as orbital-
projected density-of-states. Furthermore, their gradual transformations could
be achieved through the Si-/C-substituted graphene/silicene honeycomb lattice.

The present work is focused on the diverse geometric, electronic and magnetic
properties of C-substituted silicene. A theoretical framework, which is based
on multi-orbital hybridizations and atom-created spin configurations, is
further developed to clearly analyze the concentration- and configuration-
dependent phenomena. For example, the zero- and finite-gap behaviors, the main
features of $\pi$ and $\sigma$ bands could be understood from the specific
relations among the hybridized $\pi$, $\sigma$ and sp3 chemical bondings. On
the experimental side, a controllable synthesis way, using MTMS/hexane as
precursors, is proposed to generate the large-area graphene-based Si-C binary
2D compounds [16]. Furthermore, the silicon-carbide nanosheets are
successfully synthesized by a catalyst free carbothermal method and post-
sonication process [17], in which the AFM measurements show the average
thickness of ${\sim\,2-3}$ nm and size of ${\sim\,2}$ $\mu$m.

2\. Method of calculation

Our investigation of the diverse properties of carbon-substituted silicene is
base on density functional theory using VASP codes [18, 19]. The exchange and
correlation energies due to many-particle Coulomb interactions were calculated
with the use of the Perdew-Burke-Ernzerhof (PBE) functional under the
generalized gradient approximation [20], whereas the electron-ion interactions
can be characterized by the projector augmented wave (PAW) pseudopotentials
[21]. A plane-wave basis set with a maximum kinetic energy cutoff of 500 eV
was chosen to expand the wave function. In a direction perpendicular to the
silicene plane, a vacuum layer with a thickness of 15 $\AA$ was added to avoid
interactions between adjacent unit cells. The k-point mesh was set as $9\times
9\times 1$ in geometry optimization, $100\times 100\times 1$ for further
calculations on electronic properties via the Monkhorst-Pack scheme. During
the ionic relaxations, the maximum Hellmann-Feynman force acting on each atom
is less than $0.01$ eV/ whereas the convergent energy scale was chosen as
$10-5$ eV between two consecutive steps.

3\. Numerical Results and discussion

##  0.1 Geometric structure of carbon-substituted silicene

Carbon-substituted silicon systems are capaable of possessing unusual
geometries, as it is clearly indicated in Table 1 and Figs. 2(a) through 2(l).
Four types of typical C-substitution configurations, which cover meta, ortho,
para, and single cases, are chosen for a model investigation. In general, the
third and fourth types, respectively, possess the lowest and highest ground
state energies, i. e., the para-configuration is the most stable among them,
or it is expected to be relatively easily synthesized in experimental growths.
Carbon and silicon atoms, respectively, possess four outer orbitals of [2px,
2py, 2pz, 2s] and [3px, 3py, 3pz, 3s]. In addition to the pure ones, the
substitutions of the former, as clearly illustrated in Fig. 2(a) through 2(l),
will create the hybridized $\pi$ and $\sigma$ bondings [2pz-3pz and (2s, 2px,
2py)-(3s, 3px, 3py)] in silicene honeycomb lattice [the standard orbital
hybridizations in Fig. 2(a)]. This seems to be responsible for the extremely
non-uniform chemical/physical environments in an enlarged unit cell.
Consequently, the Si-Si/Si-C bond lengths and the height difference between A
and B sublattices might lie in specific ranges [Table 1]. Specifically, the
full C-substitution case, with the only uniform environment, is clearly
different from the pristine one, since they, respectively, have the planar and
buckled honeycomb lattices [Figs. 2(b) and 2(a); $\delta_{z}$=0 $\sim\,0.48$
$\AA$]. When the C-concentration is sufficiently low, the buckled structures
will recover, such as, the C:Si ratio lower than 15$\%$. For the other
conditions, the existence of buckling strongly depends on the concentration
and configuration of the substitution.

Figure 1: Table: The optimal geometric structures of carbon-substituted
silicene systems under various concentrations and configurations with the C-C,
C-Si and Si-Si bond lengths, the band gaps, ground state energies per unit
cell, together with the height differences between A and B sublattices. Figure
2: The top-view optimal geometric structures for C-substituted silicene
systems; the (a) pristine, (b) full-, (c) 50$\%$-meta-, (d)
$50$%-ortho-/para-, (e) 33.3$\%$-meta, (f) 33$\%$-ortho, (g) 3 3$\%$-para- (h)
12.5$\%$-meta-, (i) 12.5$\%$-ortho-, (j)12.5$\%$-para. (k) 14.3$\%$-single-
and (l) 5.8$\%$-single-substitution cases. Also shown in the narrow windows
are the side views for the buckled structures.

##  0.2 Band structure tailoring of Carbon-substituted silicene systems

The specific relations among the $\pi$, $\sigma$ and ${sp^{3}}$ bondings in a
pristine monolayer silicene account for the main features of the band
structure. The electron-hole symmetry around the Fermi level is only weakly
modified at low energy, as illustrated in Fig. 3(a). The first pair of valence
and conduction bands. being nearest to ${E_{F}=0}$, are initiated from the
stable K/K′ valley. A very narrow band gap of ${E_{g}\sim\,0.01}$ meV comes to
exist between slightly separated Dirac cones, mainly due to the weak spin-
orbital coupling [a single-particle interaction]. This result is consistent
with that obtained using the tight-binding model [22]. Additionally, they show
the valence/conduction saddle M-point structure at -1.02 eV/0.59 eV. Finally,
the $\pi$ band is ended at the stable $\Gamma$ valley, in which the $\pi$-band
energy width for valence states, the energy spacing between the initial K
point and the final $\Gamma$ point is about 3.2 eV. Apparently, such
electronic spectrum originates from the $\pi$ bonding of pure
3${p_{z}}$-3${p_{z}}$ orbital hybridizations in the buckled honeycomb lattice.
On the other hand, the four-fold degenerate $\sigma$ bands, which arise mainly
from [3px, 3$p_{y}$]-orbital bondings, obey parabolic energy dispersion
relations from the stable $\Gamma$ valley at ${E^{v}\sim\,-1.04}$ eV. The
$\sigma$ and $\pi$ valence bands have the obvious or observable mixing
behavior along the $\Gamma$K and $\Gamma$M directions, since there exists a
weak, but significant sp3 bonding. Additionally, the band width of the first
$\sigma$-electronic states along $\Gamma$MK is about 3.45 eV. Generally, all
$\pi$\- and $\sigma$-bands [three-band] widths could be well defined for
monolayer silicene. In addition, all ${\bf k}$-states in the energy bands are
doubly degenerate in terms of spin degree of freedom, since the spin-up- and
spin-down-dominated states have identical energy spectra.

Figure 3: Electronic structures, with the dominance circles of silicon and
carbon atoms [the blue and red ones], for the C-substituted silicene systems;
the (a) pristine, (b) [1:1]-, (c) [2:4]-meta-, (d) [2:4]-ortho-/para-, (e)
[2:6]-meta-, (f) [2;6]-ortho-, (g) [2:6]-para-, (h) [2:16]-meta-, (i)
[2:16]-ortho-, (j) [2:16]-para-, (k) [1:7]-single-, (l) [1:17]-para
substitution configurations.

The full carbon-substitution silicene system, i.e., the silicon-carbon binary
compound [23], exhibits an unusual electronic energy spectrum as shown in Fig.
3(b), being totally different from the pristine case [Fig. 3(a)]. The occupied
valence bands are highly asymmetric to the unoccupied conduction bands near
${E_{F}=0}$. Such material is classified as a direct-gap insulator with a band
gap larger than 2.56 eV at the K/K′ point, as observed in a wide-gap monolayer
boron-nitride system [24]. This is closely related to the strongly modified
Dirac-cone structure across the Fermi level with highly anisotropic enegy
dispersions [e.g., the partially flat conduction band along KM]. Both valence
and conduction bands nearest to $E_{F}$ are, respectively, $\pi$ and
$\pi^{\ast}$ bands due to the ${2p_{z}}$-${3p_{z}}$ impure orbital
hybridizations in Si-C bonds. The higher/lower ionization energy of 2pz/3pz
orbitals further leads to the creation of a band gap for the honeycomb
lattice, and the C-/Si-dominance of the $\pi$/${\pi^{\ast}}$ bands [the red
and blue circles]. The first pair of energy bands are highly anisotropic. For
example, the valence/conduction $\pi$/${\pi^{\ast}}$ states, which are
initiated from the K valley, show different group velocities along the KM and
K$\Gamma$ directions, especially for the saddle point/the partially flat
dispersion related to the M point. Furthermore, the $\pi$-band energy width is
${E_{w}\sim\,3.71}$ eV] according to the difference between the K- and
$\Gamma$-point energies [${-1.50}$ eV and ${-5.20}$ eV. Compared with a
pristine 3pz-3pz case [${E_{w}\sim\,3.20}$ eV in Fig. 3(a)], the larger band
width might indicate the stronger 2pz-3pz bonding. The two $\sigma$ bands,
which arise from [3px, 3py]-[2px, 2py] hybridizations, appear at
${E^{v}\sim\,-2.60}$ eV initiated from the $\Gamma$ point, but remain fourfold
degenerate electronic states there. Furthermore, the contributions from the
different orbitals are comparable in the [C, Si]-co-dominated energy bands.
The concave-downward parabolic valley near the $\Gamma$ point is dramatically
transformed into cancave-upward ones/the saddle-point forms along the
$\Gamma$K/$\Gamma$M direction. Consequently, the first $\sigma$-bandwidth is
approximately $4.30$ eV. There exist direct crossings of the $\pi$ and
$\sigma$ valence bands along any direction, e.g., M$\Gamma$ and K$\Gamma$.
This result clearly reveals no evidence of sp3 bonding [25]. As a result, the
fundamental properties of the [1:1] Si-C compound are dominated by the well-
behaved $\pi$ and $\sigma$ bondings in the absence of sp3 ones.

Figure 4: The spatial charge densities of the carbon-substituted silicene
materials for the various concentrations and configurations: (a) the absence
of guest atoms, (b) [1:1], (c) [2:4]-meta, (d) [2:4]-ortho-/para, (e)
[2:6]-meta, (f) [2;6]-ortho, (g) [2:6]-para and (h) [1:7] conditions. They are
shown on the ${[x,z]}$ and ${[x,y]}$ planes, clearly illustrating the $\pi$
and $\sigma$ chemical bondings and their orthogonality/non-orthogonality.

Electronic energy spectra are greatly enriched through the modulations of
concentration and distribution configuration. The [2:4]-meta and
[2:4]-ortho/para cases, as clearly displayed in Figs. 3(c) and 3(d), present
finite- and zero-gap behaviors, respectively, corresponding to the highest
occupied state and lowest one at the $\Gamma$ point [${E_{g}\sim\,0.90}$ eV]
and the gapless Dirac-cone structure along the $\Gamma$K direction [but not
$\Gamma$M]. That the low-lying valence and conduction bands are initiated from
the stable $\Gamma$ valley is associated with the zone-folding effect.
According to the pristine and full-substitution configurations [Figs. 3(a) and
3(b)], the valence and conduction bands in the energy range of
${|E^{c,v}<1.30|}$ eV, including the first pair, mainly arise from the [2pz,
3pz]-orbital hybridizations. Such bondings are responsible for the low-energy
physical properties. This result is also confirmed by the orbital-projected
density-of-states [discussed later in Fig. 5(b)]. It should be noted that the
latter [Fig. 3(d)] belongs to a zero-gap semiconductor because of the
vanishing density-of- states at the Fermi level. The low-lying energy bands
are dominated by the 3pz orbitals of silicon-host atoms because of the higher
weight. More energy subbands come to exist under the enlarged unit cells.
Therefore, the band crossings and anti-crossings would happen frequently. In
addition, it might be able to define the $\pi$-band energy widths through the
$\Gamma$K$\to$KM$\to$M$\Gamma$ direction [examinations from the 3pz-projected
density-of-states in Figs. 5(c) and 5(d)], respectively, corresponding to
$4.01$ eV and $4.70$ eV for the meta- and ortho-configurations. It would be
very difficultm or even meaningless, in characterizing band widths with
further decrease of guest-atom concentration, as a result of more complicated
valence subbbands. Concerning the first $\sigma$ valence bands, they are
roughly identified from the initial $\Gamma$-states at ${E^{v}\sim\,-1.80}$ eV
for the meta-case [Fig. 3(c)], while the very strong zone-folding effects
forbid their characterizations under the ortho-condition [Fig. 3(d)].

Very interestingly, two kinds of band properties are also revealed in the
[2:6] cases, as clearly indicated in Figs. 3(e)-3(g). The meta-, ortho- and
para-configurations, respectively, exhibit the small-, narrow- and zero-gap
behaviors [the finite- and zero-gap semiconductors] according to the first
pair of energy bands nearest to the Fermi level. The first case has an
indirect band gap of ${E_{g}\sim\,0.31}$ eV, being determined by the specific
energy spacing between the highest occupied state at the K point and the
lowest unoccupied state at the $\Gamma$ point [Fig. 3(e)]. Furthermore, the
first valence band has an oscillatory energy dispersions. As a result, two
stable valleys dominate low-energy physical phenomena simultaneously. The
significant overlaps of valence and conduction bands, which come to exist
along any wave vector direction, appear under the second configuration in Fig.
3(f). There are five/three energy subbands, with weak oscillatory dispersion
relations near the $\Gamma$ point/ the K and M points. They are responsible
for the formation of partially flat bands and thus the high density-of-states
near the Fermi level [discussed later in Fig. 5(f)]. The third case in Fig.
3(g) shows a gapless Dirac cone structure at the K/K′ valley only with a
single Fermi momentum state [a zero density-of-states at $E_{F}$ in Fig.
5(g)]]. There are more finite energy spacings between valence and conduction
subbands at the $\Gamma$ valley. For any guest-atom configurations, most
electronic energy spectra are dominated by the silicon atoms, while the
opposite is true for the carbon atoms.

Figure 5:  The various van Hove singularities in atom- and orbital-decomposed
density-of-states for carbon-substituted silicene systems under the various
cases: (a) pristine, (b) [1:1]-, (c) [2:4]-meta-, (d) [2:4]-ortho-/para-, (e)
[2:6]-meta-, (f) [2;6]-ortho-, (g) [2:6]-para-, (h) [2:16]-meta-, (i)
[2:16]-ortho-, (j) [2:16]-para-, (k) [1:7]-single-, and (l) [1:17]-single-
configurations, being, respectively, illustrated by the solid/dashed [red,
green, blue and purple] curves for the [3s, 3px, 3py, 3pz] orbitals/[2s, 2px,
2py, 2pz] contributions.

Clearly, there appear more complex electronic energy spectra when the guest-
atom concentrations are decreased [under the reduced/enhanced number of Si-C
and C-C/Si-Si bonds]. Typically, three sorts of distribution configurations
are revealed under the specific [2:16] condition, the meta-, ortho-, and para-
ones in Figs. 3(h) through 3(j). Their low-lying band structures exhibit
strongly modified Dirac cones, respectively, with a direct gap of
${E_{g}\sim\,0.14}$eV due to parabolic valence and conduction dispersion
relations near the $\Gamma$ valley, the gapless and crossing structure [the
finite energy spacing] along the $\Gamma$M direction [the $\Gamma$K
direction], and similar zero-gap behavior. Additionally, the single [1:7]
substitution in Fig. 3(l) creates an ${E_{g}\sim\,0.09}$ eV indirect band gap
associated with the valence K state and conduction $\Gamma$ state. It should
be emphasized that the second and third electronic structures only belong to
zero-gap semiconductors [the vanishing density-of-states at $E_{F}$ in Figs.
5(i) and 5(j)]. The Moiré superlattices have created a substantial number of
energy subbands within a smaller first Brillouin zones, leading to frequent
band crossings and anti-crossings. As a consequence, it is very difficult to
characterize the $\pi$-band width [similarly for the $\sigma$ bandwidths.
However, the orbital-projected density-of-states might be useful in examining
the relationship among the $\pi$, $\sigma$ and sp3 bondings. Interestingly,
C-substituted silicene will gradually recover to the pristine case [Fig. 3(a)]
at very low guest-atom concentrations, e.g., the appearance of an almost
gapless Dirac cone under the [1:17] case in Fig. 3(l). In summary, the energy
subbands, with ${|E^{c,v}<1.5|}$ eV, are dominated by the [2pz, 3pz] orbital
hybridization, and the other deeper-/higher-energy electronic states might be
closely related to the [2px, 2py, 2s, 3px, 3py 3s] or eight orbitals in Si and
C atoms. The carbon substitutions in silicene can only induce the modified
Dirac cone structures, but not free carriers [the $n$\- or $p$-type dopings].

High-resolution angle-resolved photoemission spectroscopy (ARPES) measurements
provide a convenient way for examining/identifying occupied electronic energy
spectra along specific 2D/1D wave vector directions. This technique has
identified several unusual low-lying energy dispersion relations in graphene-
related sp2-bonding materials, such as, the linear valence Dirac cone in
monolayer graphene [26], two pairs of parabolic bands in bilayer AB stacking
[27], partially flat, sombrero-shape and linear energy bands in trilayer ABC
stacking [28], bilayer-/monolayer-like energy bands of Bernal graphite at the
K/H corners [AB-stacked one at ${k_{z}=0}$/${k_{z}=\pi}$; [29], and 1D
parabolic energy subbands of graphene nanoribbons [30], The delicate and
thorough ARPES examinations are required for the predicted band structures of
C-substituted silicene materials. That is, they cover the zero- and finite-gap
behaviors with the modified Dirac-cone structures across the Fermi
level/without the carrier dopings, the $\pi$ band/subbands with the initial K
or $\Gamma$ valley, the intermediate saddle-M-point form $\&$ the final
$\Gamma$ valley, the degenerate $\sigma$ bands initiated at deeper energies
from the $\Gamma$ valley, their band crossings $\&$ anti-crossings along the
K$\Gamma$ and M$\Gamma$ directions, and spin-degenerate electronic states,
being very sensitive to the configuration and concentration of guest atoms
[Figs. 3(a) through 3(l)]. The full information about the verified band
structures is very useful in understanding the substitution-enriched relations
among the $\pi$, $\sigma$ and sp3 bondings in Si-Si, Si-C and C-C chemical
bonds, and the non-magnetic property.

The first-principles calculations acknowledge that the tight-binding model is
capable of simulating the low-lying spin-degenerate energy dispersions in
carbon-substituted silicene systems. The site energies of different orbitals,
the impure/pure multi-orbital hybridizations, the non-uniform environments
[the optimal positions of host and guest atoms], and the buckled/planar
structures need to be taken into account simultaneously. For example, the full
C-substitution, which covers the uniform Si-C bonds in the smallest unit cell,
possesses the distinct orbital energies of [3s, 3px, 2s, 2px] and the impure
$\pi$ [(3pz-2pz)] and $\sigma$ [(3s, 3px, 3py)-(2s, 2px, 2py)] chemical
bondings. In general, it is difficult to determine the various hopping
integrals due to the multi-orbital hybridizations, especially for those of
Moiré superlattices with many atoms in large unit cells. Maybe, Harrison’s
rule [31] is an efficient method for solving the non-uniform chemical bonds
under dilute substitutions. When the suitable parameters are achieved under
numerical fitting with the first priciples calculations, the generalized
tight-binding models in the presence of external electric and magnetic
fields[32, 33] are very powerful in completely understanding the rich magnetic
quantization phenomena of Si-C-based 2D compounds. The similar simulation
methods could be done for the more complicated B- and N-substituted silicene
compounds.

##  0.3 Spatial charge density distributions

The spatial charge density distributions are able to provide certain
observable evidences regarding the existence of the impure/pure $\pi$ and
$\sigma$ chemical bondings as well as their non-orthogonalities or
orthogonalities. Pristine 2D silicene, as clearly displayed in Fig. 4(a) with
the ${[x,z]}$\- and ${[x,y]}$-plane projections, shows the well-defined
${3p_{z}}$-${3p_{z}}$ and ${[3s,3p_{x},3p_{y}]}$-${[3s,3p_{x},3p_{y}]}$
orbital hybridizations, especially for the latter. In our notation, $\rho$ is
a symmetric distribution around the Si-Si bond center. Most of the charge
density is accumulated between two silicon atoms [the red region],
corresponding to the very strong $\sigma$ bonding of three orbitals. The
neighboring Si-atoms are also attracted together through the parallel
${3p_{z}}$ orbitals perpendicular to the ${[x,y]}$ plane, in which the $\pi$
bonding appears in the outer region shown by the light green/blue color. As a
result of buckling, they have the weak, but significant ${sp^{3}}$
hybridizations under the non-orthogonality of $\pi$ and $\sigma$ chemical
bondings. When the Si-Si bonds become Si-C ones under full substitution,
$\rho$ presents a dramatic transformation, being clearly illustrated in Fig.
4(b). The charge density is highly asymmetric with respect to the C-Si bond
center, mainly owing to the different electron affinities of the guest and
host atoms. There exist more carriers around the guest C-atoms [the red
region]; that is, electrons are transferred from silicon to carbon atoms. The
whole C-Si bonds consist of a planar honeycomb lattice with a stronger
$\sigma$ bonding, compared with those of a pristine one [Fig. 4(a)]. The
$\sigma$ and $\pi$ bondings could be roughly defined under the impure
${2p_{z}}$-${3p_{z}}$ and ${[2s,2p_{x},2p_{y}]}$-${[3s,3p_{x},3p_{y}]}$
orbital hybridizations, respectively. They are directly reflected in the
$\pi$\- and $\sigma$-electronic valence subbands, with direct crossings [Fig.
3(b)]. Most important, two kinds of chemical bondings are orthogonal to each
other. Therefore, the ${sp^{3}}$ orbital hybridizations are negligible in the
[1:1] case. Each Si-C bond has the identical chemical environment, i. e., only
one chemical bond in a unit cell. This clearly illustrates that it is
relatively easy to simulate the first-principles band structure using the
tight-binding model [34].

When the carbon concentrations are decreased [Figs. 4(c) through 4(h)], the
existence of $\pi$ and $\sigma$ chemical bondings is relatively easily
examined from the spatial charge densities. There exist C-Si, Si-Si and even
C-C bonds, for which the last ones are stable under the specific ortho cases
[Table 1]. The almost symmetric carrier distributions are clearly revealed
between two silicon atoms for any concentration and configuration, in which
they possess the lowest charge density among three kinds of chemical bonds.
Similar phenomena appear for C-C bonds with the highest carrier densities,
e.g., $\rho$ values in Figs. 4(d) and 4(f) at ${[2:4]}$ and ${[2:6]}$ ortho
conditions, respectively. Roughly speaking, the spatial charge densities in
different chemical bonds are not sensitive to changes in various carbon
substitutions. These further illustrates that the $\sigma$ and $\pi$ chemical
bondings might be well separated from each other. Therefore, they could be
roughly defined in the carbon-substituted silicene systems. As a result, the
fundamental low-energy properties are expected to be dominated by the
$\pi$-electronic states due to the modified Dirac cone structures. On the
other hand, the different chemical bonds lead to complex orbital
hybridizations and thus contribute to the difficulties in obtaining suitable
phenomelogical models. When the first principles electronic energy spectra
along the high-symmetry paths are successfully simulated by the tight-binding
model with the non-uniform and multi-/single-orbital hopping integrals [35],
the diversified essential properties could be fully explored in the near-
future, e.g., the rich and unique magnetic quantization phenomena [33], as
predicted/observed in layered graphene systems [36].

##  0.4 Density-of-states

There are four/five categories of van Hove singularities in carbon-substituted
silicene systems or pristine one, as clearly illustrated in Figs. 5(a) through
5(f). The orbital- and atom-decomposed density-of-states are very useful in
fully understanding the bonding-induced special structures. The critical
points, i.e., the band-edge states, in the energy-wave vector space include
the linear Dirac cone structure, local minima or maxima of parabolic energy
dispersion, saddle points, constant energy loops, which are closely related to
band anticrossings, and partially flat bands [Figs. 3(a) through 3(l)]. Their
densities-of-states generate the V-shape form, discontinuous shoulders,
logarithmic divergent peaks, asymmetric peaks in the square root divergence,
and delta-function-like peaks, respectively. A pure monolayer silicene in Fig.
5(a) displays an almost linear $E$-dependence across the Fermi energy with a
vanishing density-of-states [a quasi-V shape in the range ${-0.5}$
meV${\leq\,E\leq 0.5}$ meV; the purple curve of 3pz orbitals], a logarithmic
symmetric $\pi$-peak/${\pi^{\ast}}$-peak at $-$1.02 eV/0.59 eV, the [$\pi$,
$\sigma$]-mixing created square root asymmetric peaks at [$-$2.20 eV, $-$2.65
eV, $-$3.15 eV. The red, green and blue curves of (3px, 3px, 3py) orbitals],
the $\Gamma$-valley $\pi$ shoulder at $-$3.20 eV, the initial $\sigma$-[3px,
3py] shoulder at ${-1.03}$ eV at the $\Gamma$ point, their M saddle point
symmetric peak at ${-2.47}$ eV and the K valley shoulder at ${-4.40}$ eV.
Consequently, the $\pi$ and $\sigma$ bandwidths are $\sim$3.15 eV and 3.36 eV,
respectively, It should be noted that the 3s orbitals [the red curve] are
frequently accompanied by [3px, 3py] ones, but their contributions become
significant at the deeper/higher energies, e.g., density-of-states within
${E<-3}$ eV. The above mentioed features of van Hove singularities further
illustrate the well behaved $\pi$ and $\sigma$ chemical bondings and their
weak, but important hybridization.

The structural features, energy and number of van Hove singularities exhibit a
dramatic transformation under full carbon substitution, as indicated in Fig.
5(b). The special structures cover vanishing density-of-states within a band
gap of ${E_{g}\sim\,2.56}$ eV at the K/K′ valley, the initial $\pi$
shoulder/delta function-like $\pi^{\ast}$ peak at $-$1.26 eV/1.26 eV [also
shows the low-lying $\pi$ and $\pi^{\ast}$ bands in Fig. 3(b)], the $-$1.95 eV
symmetric $\pi$ peak in the logarithmic divergence, the final $\pi$ shoulder
at $-$5.00 eV, the first $\sigma$ shoulder at $-$2.75 eV, the $-$4.10 eV
logarithmic peak, and the second discontinuous structure at $-$7.01 eV. We
also, observe that the first valence/conduction structure is dominated by the
C-2pz orbitals/Si-3pz ones due to an obvious difference of ionization energy.
The widths of the $\pi$ and first $\sigma$ bands are $\sim$3.74 eV and 4.26
eV, respectively. Additionally, there is no evidence of $\pi$-$\sigma$ band
mixing. This is mainly due to the absence of the simultaneous four-orbital
structures. The fact that the sps bonding is absent agrees with the direct
$\pi$-$\sigma$ subband crossings [Figs. 3(b)].

Within the whole energy range, the van Hove singularities become more complex
during the decrease of guest-C-atom concentration, as a result of zone-folding
effects as well as the significant [3s, 3px, 3py, 3pz]-[2s, 2px, 2py, 2pz]
multi-orbital hybridizations. For example, the [2;4]-meta and [2:4]-ortho
configurations, as clearly illustrated in Figs. 5(c) and 5(d), exhibit diverse
low-energy van Hove singularities arising from the dominant 3pz-2pz chemical
bonding. The former has a $\pi$-electronic zero density-of-states in the 0.90
eV-gap region [the $\Gamma$ valley in Fig. 3(c)], a threshold valence shoulder
at ${-0.45}$ eV [the first composite conduction state structure due to the
discontinuous shoulder and delta function-like peak at 0.45 eV], a strong
logarithmic peak at ${-1.41}$ eV, as well as the second and third step
structures at ${-1.69}$ eV and ${-2.71}$ eV, respectively. Regarding the
latter, the 3pz-2pz-diversified van Hove singularities show a gapless V-shape
[a Dirac cone energy spectrum in Fig. 3(d)], the first valence shoulder at
${-0.5}$ eV [the initial conduction shoulder at 0.30 eV], the logarithm-step
composite structure at ${-1.01}$ eV, the symmetric peak arising from two
opposite shoulders at ${-1.32}$ eV, a similar one at ${-2.10}$ eV, and the
${-2.65}$-eV asymmetric peak in the square root form. The widths of the
$\pi$-band in the former and latter cases are estimated to be $\sim$4.05 eV
and 4.70 eV, respectively. Furthermore, the first $\sigma$-bands are wider
than 4.5 eV. Obviously, the low-energy physical phenomena are dominated by the
3pz orbitals of Si-host atoms.

As the guest-atom concentration is declined, there appear additional van Hove
singularities. Overall, in this way, one is able to characterize the width of
the $\pi$-band from the effective distributions of the [3pz, 2pz] orbitals for
any concentration and configuration [Figs. 5(a) through 5(l)]. However, the
opposite is true for the first $\sigma$-band width except for full
substitution and the pristine cases [Figs. 5(a) and 5(b)]. Certain van Hove
singularities, close to the Fermi level, are very useful to comprehend the
band-edge states in the first pair of the valence and conduction bands. The
main features cover the vanishing density-of-states at/across the Fermi level
[Figs. 5(c) through 5(e)]/[Figs. 5(d) through 5(g)], the electron-hole
asymmetry near the first pair of clearly identifiable shoulders, the prominent
$\pi$ peak in logarithmic form, the second and third shoulders and so on.
Finally, the dip structure at $E_{F}$, with a very small gap comparable to the
broadening factor, will come to exist for a sufficiently low concentration,
e.g., the almost gapless behavior in the [1:17] case [Fig. 5(l)]. We also
noticed that the single-particle interactions of spin-orbital couplings cannot
create any significant effects on the electronic properties and thus the other
essential properties.

High-resolution scanning tunneling spectroscopy (STS) is an efficient method
which could serve to identify the valence and conduction band-edge states [the
van Hove singularities of the density-of-states. So far, they have confirmed
the rich and unique density-of-states in dimension, stacking, layer number
dependent graphene based materials, such as, the width and edge structure
dominated/chirality and radius related band gaps and considerable asymmetric
prominent peaks in 1D semiconducting graphene nanoribbons/carbon nanotubes
[37]/[38], the V-shape structure with a dip at the Fermi level in monolayer
graphene [39], the Dirac point red shift thorough alkali adatom chemisorption
[$n$-type doping] [40], the gate-voltage-created band gap in bilayer AB-
stacked graphene [41], surface state peak just at $E_{F}$ in trilayer ABC
stacked graphene [42], the semimetallic behavior [finite density-of-states at
$E_{F}$] and the clearly symmetric $\pi$ and ${\pi^{\ast}}$ peaks at mid
energies for Bernal graphite [43]. Similar STS examinations could be conducted
on the various van Hove singularities in carbon-substituted silicene systems,
covering the zero, finite gap $\&$ semimetallic band properties, the $\pi$ and
$\sigma$ band widths, and the configuration as well as concentration dependent
five types of special structures [V shape across the Fermi level, shoulders,
symmetric peaks in the logarithmic form, delta function-like ones, i.e.,
square root asymmetric peaks in Figs. 5(a) through 5(l)]. Both experimental
STS and ARPES results would provide sufficient information on the strength
relations among the $\pi$, $\sigma$ and sp3 bondings during the chemical
substitutions of guest atoms.

4\. Concluding Remarks and Summary

Chemisorption and substitution of Si-guest atoms on monolayer graphene present
unusual geometric properties, which are directly reflected in the spatial
charge distributions. Si-adsorbed graphene is a non-buckled honeycomb lattice
under optimal bridge-site positioning. A planar structure clearly indicates a
very small variation in the $\sigma$ bonding of C-[2s, 2px, 2py] orbitals.
Therefore, they hardly take part in Si-C bonds. This adsorption configuration
is similar to that of graphene oxide [44], but there is a slight buckling in
the latter. The Si-C bond length is about 2.1-2.5 $\AA$, leading to the
significant multi-orbital hybridizations. Furthermore, the C-C bond lengths
remain almost unchanged. According to the detailed analyses on the 3D charge
density distributions, their spatial variations, as well as the atom and
orbital projected density-of-states, the sp3-p multi-orbital hybridizations
are deduced to determine the chemical Si-C bonds.

The theoretical predictions on the bridge site and Si-C/C-C bond lengths could
be examined using high-resolution STM/TEM/LEED measurements. Also, Si-
substituted graphene systems display planar geometries under multi-orbital
chemical bondings in Si-C/C-C/Si-Si bonds. The Si-C bond length is about
1.62-1.83 $\AA$, being shorter than those in the Si-adsorbed systems. This
means that four orbitals in host and guest atoms are strongly hybridized with
one another. The STM experiments are suitable for examining the predicted
Si-C, C-C and Si-Si bond lengths. Any chemisorption and substitution of Si-
guest atoms significantly modify the unusual band structure of monolayer
graphene, especially the zero gap semiconducting behavior and linear Dirac
cone due to the $\pi$ bondings of the 2pz orbitals. In the 100$\%$ double- and
single-side adsorption cases, Si-adsorbed graphene systems are semimetals with
free conduction electrons and valence holes. For the low concentration
systems, they belong to the $p$-type metals with only free valence holes. In
general, the Dirac cone structure near the stable valley is seriously
distorted after Si-chemisorption. There are more valence and conduction energy
subbands, accompanied by the various band-edge states within an entire energy
spectrum, e.g., the emergent low-lying energy bands along the K$\Gamma$ and
M$\Gamma$ directions. However, the $\sigma$ bands, which arises from the [2px,
2py] orbitals of carbon atoms, exhibit a robust red shift of $\sim$1 eV and
$\sim$0.5 eV for 100$\%$ and lower-concentration cases, respectively.

The above mentioned important results directly reveal critical mechanisms,
i.e., , the multi-orbital hybridizations of sp3-p in Si-C bonds, sp3-sp3 in
Si-Si bonds, and sp2-sp2/$\pi$ in C-C bonds. Such chemical bondings mainly
produce four Si- and C-orbitals, as a result of the bridge-site adsorption
positions, as well as the binding energies and charge distributions of the
separated orbitals. High resolution ARPES measurements are very useful in
verifying the low-lying valence bands near/crossing the Fermi level along
$\Gamma$K and $\Gamma$M and the rigid $\sigma$ bands initiated from the
$\Gamma$ valley. On the other hand, all substitution results in semiconducting
behavior with a finite or zero band gap. The Dirac cone structure presents a
deviation from the $\Gamma$ point, a strong distortion, or even destruction.
The number of valence and conduction energy subbands remains the same after
chemical substitutions. Furthermore, they are co-dominated by the Si-guest and
C-host atoms.

The main features of the band structure in Si-substituted graphene are
dominated by the sp2-sp2 and p-p orbital hybridizations in Si-C/C-C/Si-Si
bonds. Several unusual van Hove singularities in the atom and orbital
decomposed density-of-states are created under the Si-guest-atom chemisorption
and substitution. Pristine monolayer graphene only presents a V-shaped
structure with a vanishing value at the Fermi level, the logarithmic symmetric
$\pi$-/${\pi^{\ast}}$-peaks at $-$2.41 eV/1.81 eV, and a shoulder structure at
${-3.01}$ eV, in which the former two and the last one, respectively,
correspond to the $\pi$ and $\sigma$ bondings of carbon atoms. The strong
$\pi$-bonding evidence is totally destroyed by chemisorption except for the
very diluted Si-adatoms. They are replaced by a finite density-of-states at
${E_{F}=0}$, many shoulders, and peak structures. Furthermore, they are co-
dominated by the four Si-[3s, 3px, 3py, 3pz] orbitals and the single C-2pz
orbital, since their contributions are merged together. Specifically, the
$\sigma$-band shoulder comes to exist at $-$4.21 eV, $-$4.11 eV or $-$3.5 eV,
and the $\sim$0.5-1.0 the red shift is closely related to the distinct
ionization energies of Si and C atoms. The above-mentioned significant
features further support and illustrate the sp3-sp3 $\&$ sp2-sp2 multi-orbital
hybridizations in Si-C bonds and C-C bonds, respectively. Concerning the
substitution cases, density-of-states is zero within a finite energy range
centered at the Fermi level, except for few systems with deformed V-shaped
structures at $E_{F}$ . All the C-substituted silicene systems are finite- or
zero gap semiconductors. The special structures, which originate from Si-3pz
and C-2pz orbitals, appear simultaneously. Also, a similar behavior is
revealed in Si-[3px, 3py] and C-[2px, 2py] orbitals. These further illustrate
the critical single- and multi-orbital hybridizations in Si-C bonds. The
predicted van Hove singularities could be verified by the high-resolution STS
experiments [45].

Acknowledgments

This work was financially supported by the Hierarchical Green-Energy Materials
(Hi-GEM) Research Center, from The Featured Areas Research Center Program
within the framework of the Higher Education Sprout Project by the Ministry of
Education (MOE) and the Ministry of Science and Technology (MOST
108-3017-F-006 -003) in Taiwan.

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Table Captions

Table 9.1: The optimal geometric structures of the carbon-substituted silicene
systems under the various concentrations and configurations with the C-C, C-Si
$\&$ Si-Si bond lengths, the band gaps, the ground state energies per unit
cell, and the height differences between A and B sublattices.