# Topological viewpoint of two-dimensional group III-V and IV-IV compounds in
the presence of electric field and spin-orbit coupling by density functional
theory and tight-binding model

A. Baradaran<EMAIL_ADDRESS>M. Ghaffarian  Department of Physics,
University of Qom, Qom, Iran

###### Abstract

Using the tight-binding model and density functional theory, the topological
invariant of the two-dimensional (2D) group III-V and IV-IV compounds are
studied in the absence and the presence of an external perpendicular electric
field and spin-orbit coupling. It will be recognized that a critical value of
these parameters changes the topological invariant of 2D graphene-like
compounds. The significant effects of an external electric field and spin-
orbit coupling are considered to the two-center overlap integrals of the
Slater-Koster model involved in band structures, changing band-gap, and tuning
the topological phase transition between ordinary and quantum spin Hall
regime. These declare the good consistency between two theories: tight-binding
and density functional. So, this study reveals topological phase transition in
these materials. Our finding paves a way to extend an effective Hamiltonian,
and may instantly clear some computation aspects of the study in the field of
spintronic based on the first-principles methods.

###### keywords:

quantum spin Hall effect , tight-binding approximation , density functional
theory.

###### MSC:

[2010] 81-08 , 81S99

##  1 Introduction

Topological insulators (TIs) have been considered in much researches in the
last decade [1, 2, 3, 4]. This innovative material is a nontrivial insulator
phase that is insulating in bulk but holds stable metallic edge states, which
are protected by time-reversal symmetry (TRS) and spin-orbit interaction (SOI)
[5, 6]. The two-dimensional (2D) TIs, which are known as quantum spin Hall
(QSH) insulators, are suitable for the novel device because of robust edge
states against backscattering caused by impurities [7, 8, 9].

Since the discovery of graphene [10], graphene-like materials have achieved
attention due to their familiar 2D honeycomb structure. Some of these
materials have stable flat structures exactly like graphene, such as single-
layer of boron pnictides [11], and others are stable in buckled geometry, such
as silicene [12, 13], germanene [13] and stanene [14]. In imitation of the
discovery of the TI phase in graphene as 2D Kane-Mele material induced SOI
[5], the buckled monolayers of the group III-V and IV-IV such as monolayer of
Si, Ge, and Sn [15]; GaAs film by tensile strain [16]; Bi and Sb quasi-two
dimensions [17, 18], and arsenene [19] allow us to observe the QSH. Moreover,
SOI in 2D materials has been known in two ways, intrinsic; and Rashba spin-
orbit that can be controlled by a perpendicular electric field which is
generally smaller than the intrinsic SOI [20, 21]. The effect of these
parameters on the QSH phase needs more exploring by tuning SOI or applying an
EF. Density functional theory and tight-binding (TB) model [22, 23] provide us
suitable tools to study these effects. In particular, 2D monolayer compounds
have been of precise interest to us, whose topological behavior can be
investigated with the simplest TB model [24, 25] as the nearest neighbor
atoms, with SOI [26] or in the presence of an EF [27]. These researches
inspiring for us to investigate a more comprehensive study of 2D monolayers of
the group III-V and IV-IV.

The binary monolayer structures of compounds of the groups III-V and IV-IV, in
addition to the elements of these groups, can be formed a buckled honeycomb
lattice and stabilize by cutting in the direction of (111) (Fig. 1). The
atomic structure, phonon modes, mechanical properties, and electronic
structure for some of these materials and their heterostructures have been
studied by Şahin et al. [28]. Then, others showed that the electric field
produces a tunable band-gap in Dirac-type electronic spectrum, and
investigated in silicene [29]. Also, the electronic structure of silicene was
simulated by the TB method with basis $sp^{3}d^{5}s^{*}$ [30], and the
effective Hamiltonian in the vicinity of Dirac point was constructed. Even,
Liu et al. derived the low-energy effective Hamiltonian involving SOI for
silicene and 2D germanium and tin [31]. They showed that Si and Ge monolayers,
despite their LB geometrical structure, have electronic band structures that
are similar to graphene. Also, in the buckled structures of stanene and
silicene with hybridization $sp^{3}$ type [32], the hexagonal Brillouin zone
(BZ) attributes a massless Dirac fermion character to the charge carriers.
Other attempts were made to account for the effect of spin splitting in
buckled monolayers of group-IV [33], and the effect of an electric field of
GaAs monolayer [34] from first-principles calculations. They also sought a
robust large gap 2D topological insulators in hydrogenated III-V buckled
systems [35]. However, in all these works, the monolayer of the groups III-V
and IV-IV is not considered together from a topological point of view that was
the starting point for our work.

In this research, inspired by recent developments and exploring 2D monolayer
graphene-like materials using DFT calculation, the band structures of group
III-V and IV-IV compounds are obtained, and the QSH effect is studied by
calculation $\mathbb{Z}_{2}$ from a topological viewpoint [17] by OpenMX
package [36, 37, 38, 39, 40, 41]. Since the band-gap size is effective on the
topological behavior of these materials, we show the topological properties of
them with the band-gap. It is well known the size of the band-gap in TIs
presents an urgent challenge that could be impressible by applying external
EF, tuning SOI, and strain-induced [15, 19, 29, 34]. So, we investigate the
effect of perpendicular EF and SOI on the topological phase transition of some
compounds. For well understanding of the effect of external EF and SOI on
electronic structures of these compounds, the Slater-Koster parameters of them
in TBA are extracted with fitting DFT results using fitting methods and
packages [42, 43]. Therefore, we would try to explain the TB model well
effective and in detail. The present work, which considers a total of $15$
binary compounds in a 2D honeycomb structure and reveals for some of these
materials, such as monolayer of BSb and SiGe, a phase transition has occurred.
Fitting of DFT and TB model outputs for band structures would be carried out
to find the Slater-Koster overlap integrals in the TB model, even in the
presence of a perpendicular EF. The origin of the band-gap and its magnitude,
which plays a key role in changing the topological phase of the materials, has
been carefully studied too. The work done in this paper on these materials
deals with the topic of the topological insulators and integrally examines all
the 2D monolayers of the group IV-IV and group III-V, while previous
researches have either not addressed the topic of the topological phase or
have only examined some present compounds. These efforts nourish an outline
for future research of an effective Hamiltonian and can be quite promising for
nanoelectronics. Therefore, this article provides a context for the spintronic
applications of 2D topological graphene-like materials [44] by tunning SOI or
EF.

##  2 Methods

We have performed first-principles calculations within fully relativistic DFT
at the Perdew–Burke–Ernzerhof (PBE) level [45] for relaxation, total energy,
and electronic structures via the OpenMX package [38, 39]. All calculations
were done by using the generalized gradient approximation (GGA) as the
exchange-correlation functional. Norm-conserving pseudopotentials [46] and the
linear combination of multiple pseudo-atomic orbitals were implemented for
wave function expansion [40, 41]. The cutoff radius for the various elements
used in this article, from the lightest element, B to the heaviest, Bi, is
between $2.1~{}a.u.$ and $2.8~{}a.u.$. We have set a standard choice for
pseudo-atomic orbitals [47], and 350 Ry for cutoff energy, and k-space
sampling points of $21\times 21\times 1$ for the reciprocal lattice vectors.
The calculations have incorporated the SOI by a $\mathit{j}$-dependent
pseudopotential composed relativistic (fully relativistic pseudopotential)
[48] based on the OpenMX database 2019 (PBE19) [47]. This system is considered
a single-layer slab model with a vacuum of more than five times the lattice
constant, which ignores the interaction between the layers.

Attempts to fit the conduction bands of material with the nearest-neighbor
$sp^{3}$ tigh-binding model have generally succeeded. A quantum technical
software package to construct a TB model has been used named Tight Binding
Studio [42, 43] starting from the simplified linear combination of atomic
orbitals method in combination with first-principles calculations. Also, TB
parameter fitting package (TBFIT) [43] for the Slater-Koster method has been
applied increasing the reliability and validity of the results.

We have used the methods for computing the $\mathbb{Z}_{2}$ invariant by
utilizing the lattice Chern number (LCN) [17] in the OpenMX package because
the parity method cannot be applied to systems with EF [3], or with two
different atoms in a unit cell, so the $\mathbb{Z}_{2}$ invariant have been
computed by the methods applicable without inversion symmetry [2].
$\mathbb{Z}_{2}=1$ (mod 2) corresponds to the topological insulator, and
$\mathbb{Z}_{2}=0$ (mod 2) corresponds to the trivial insulator.

##  3 Atomic structures

Graphene forms a planar (PL) honeycomb structure since it exhibits purely
$sp^{2}$ hybridization. On the other hand, some elemental honeycomb systems
that have been found so far are not planar and form a buckled structure.
Instantly, silicene, germanene, stanene, and some compounds of group-IV
elements form a buckled structure, also for group III-V materials. These 2D
honeycomb structures can be formed by cutting the bulk crystal along (111)
Miler index. These observations make it important to study if there is a
stable honeycomb geometry or not. The results of optimized geometric
structures for the materials are summarized in Table.1, as Geo. and Azimuth
degree ($\theta$).

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Figure 1:  (top) Schematic view of 2D honeycomb lattice, (bellow) buckling
parameter, $\Delta$, and Azimuth degree, $\theta$.

The relaxation procedure of atomic structures of these compounds has been
shown that almost a little more than half of these structures have been
relaxed in a planar geometry and others have relaxed in low-buckled (LB)
geometry. So, the 2D hexagonal structure of binary compounds of group-IV
elements, their binary compounds, and group III-V compounds all form a
honeycomb structure.

The 2D hexagonal lattice in which atoms have been arranged to form a PL
honeycomb structure and LB one as shown in Fig. 1. There is a significant
difference between PL and LB geometry that is the buckling parameters
$\Delta$, causes covalent $\sigma$ bonds derived from the planar hybrid
$sp^{2}$ orbitals between adjacent atoms, changed to $sp^{3}$ hybridization.
The buckling parameter has been reported in Table. 1 based on the lattice
constant, $a$. Among these materials, germanene has a maximum value of
$\Delta$ in terms of lattice constant, $a$, and other LB compounds have the
values of $\Delta$ less than $20$ ⁒ of lattice constant. Next, first nearest-
neighbor distance (b), which will be used in A, is exactly equal to
$\frac{a}{\sqrt{3}}$ in PL structures.

Table 1:  Calculated results for group-IV elements, and group III-V compounds having a 2D honeycomb structure. Stable structures are identified as PL or LB standing for the planar and low-buckled geometries, respectively. The values of buckling parameters $\Delta$; 2D hexagonal lattice constant, $|\vec{a_{1}}|=|\vec{a_{2}}|=a$, are given. Some of the structural parameters are illustrated in Fig. 1. |  | $a$ | $b$ | $\Delta$ | band-gap$(eV)$ | $\theta$ |  $v_{F}$(FP) | TI
---|---|---|---|---|---|---|---|---
| Geo. | $(\text{\AA})$ | $(\text{\AA})$ |  | no SOI | with SOI | degree |  $\times 10^{5}$$(m/s)$ | (init.)
| Grp. | III-V |  |  |  |  |  |  |
BN | PL | 2.51 | 1.45 | - | 4.66 | 4.66 | 90.00 | 0.102 | no
BP | PL | 3.18 | 1.83 | - | 0.88 | 0.88 | 90.00 | 0.053 | no
BAs | PL | 3.36 | 1.94 | - | 0.76 | 0.76 | 89.99 | 0.014 | no
BSb | PL | 3.69 | 2.13 | - | 0.37 | 0.36 | 90.00 | 0.050 | no
BBi | LB | 3.82 | 2.20 | 0.12$a$ | 0.50 | 0.47 | 101.39 | 0.058 | no
| Grp. | IV |  |  |  |  |  |  |
Graphene | PL | 2.46 | 1.42 | - | 0.00 | 0.00 | 90.00 | 8.31 | yes
Silicene | LB | 3.80 | 2.19 | 0.19$a$ | 0.00 | 0.00 | 108.42 | 5.06 | yes
Germanene | LB | 3.93 | 2.27 | 0.20$a$ | 0.02 | 0.02 | 109.63 | 4.68 | yes
Stanene | LB | 4.54 | 2.62 | 0.19$a$ | 0.00 | 0.07 | 108.46 | 4.85 | yes
SiC | PL | 3.12 | 1.80 | - | 2.40 | 2.40 | 90.00 | 0.12 | no
SiGe | LB | 3.86 | 2.23 | 0.16$a$ | 0.01 | 0.02 | 105.55 | 3.64 | yes
GeC | PL | 3.19 | 1.84 | - | 2.08 | 2.13 | 89.96 | 0.15 | no
SnC | PL | 3.61 | 2.08 | - | 1.69 | 1.68 | 89.97 | 0.11 | no
SnSi | LB | 4.21 | 2.43 | 0.17$a$ | 0.25 | 0.23 | 106.88 | 0.42 | no
SnGe | LB | 4.28 | 2.47 | 0.19$a$ | 0.23 | 0.19 | 107.89 | 0.49 | no

Some aspects of the stability of these materials such as phonon analysis had
been studied in Ref. [28]. We also report the carrier Fermi velocity ($v_{F}$)
around the Dirac point K from first-principles with linear approximation,
which is comparable with the results of Ref. [31].

##  4  Electronic structures

Band structures of group III-V compounds have been shown in Fig. 2(a),
exhibited all electronic bands have a band-gap more than $0.4~{}eV$, but BSb
2D honeycomb lattice has the least band-gap, a little less than $0.4~{}eV$.
The authors know that DFT calculation based on GGA approximation
underestimates band-gap [49], however, the pattern of change in this value is
important for the appearance of the topological phase. In the group-IV
materials (Fig. 2 (b),(c))) those which contain a Dirac point at $K$ point are
like Graphene, and others have band-gap values from nearly $0.25~{}eV$ in SnSi
and SnGe 2D monolayers to more than $2~{}eV$ in SiC, GeC, and SnC 2D honeycomb
structures. Germanene has the highest buckled parameter, and like group IV-IV
crossed structures, the Dirac point is exactly settled on the fermi level
despite earlier report [13]. All of the band structures have been derived
without considering SOI, then when we want to mention the topological phase,
need to explore energy bands considering SOI.

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Figure 2: (color online). DFT band structure of monolayer of (a) BX (X=N, P,
As, Sb, Bi), (b) group IV crossed (zero band-gap), and (c) group IV with non-
zero band-gap; inset figures show the band near Fermi level around $K$ high
symmetry point within a narrower energy window.

When the calculations include SOI, band structures modify in the compounds
with heavier elements, like antimony (Sb), bismuth (Bi) as shown at Fig. 3(a),
and tin (Sn) has been exhibited at Fig. 3(b), (c). As we have expected SOI is
well strong in BSb and BBi honeycomb structures, therefore, in the BX (X=N, P,
As, Sb, Bi) honeycomb lattices, these structures are candidates for TI phase.
So, we selected BSb monolayer structure, especially due to PL geometry
relative to another candidate, so the discussion about Rashba SOI will be
omitted [31]. Similarly, In the group III-V compounds, the monolayer of
stanene and SiGe seems more appropriate due to larger SOI effects.

|
---
|
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Figure 3: (color online). DFT band structure in presence of SOI for a
monolayer of (a) BX (X=N, P, As, Sb, Bi), (b) group-IV crossed (zero band-
gap), and (c) group-IV with non-zero band-gap; inset figures show the band
near Fermi level around K high symmetry point within a narrower energy window.

As the Fig. 3 shows, the effective SOI in LB geometry opens a band-gap at the
Dirac points and establishes the QSH effect. The values of band-gap with and
without SOI summarize in Table 1. We refer to this discussion later.

##  5  Topological view point via first-principles calculations

In 2D structures, by changing some physical parameters such as external EF
perpendicular to the lattice surface, and SOI strength, a Dirac point appear
near the Fermi surface [27], and we expect a non-trivial topological phase. In
the following, we consider some structures from Table 1 based on the ability
to close the band-gap under changing these physical parameters.

Now, we try to classify some candidates with narrower band-gap (lower than 0.5
eV) for TI phase. There are some nominees such as BSb, SiGe, SnSi, and SnGe.
In the following, we will examine the energy band structure in the presence of
spin-orbit interaction and electric field.

In the case of the monolayer of BSb when the strength of SOI changes,
topological effects influence the band structure, and the band-gap tends to
close, so trivial insulator changes to TI. Computing $\mathbb{Z}_{2}$
invariant, by using LCNs, as shown at Fig. 4 (b), (c), confirms that a phase
transition occurred near a critical SOI strength. As this is depicted, the
band-gap descends down slowly in $4.2$ times the initial strength of SOI,
Table 2 gives the band-gap values. In this table, for instance, $\times 2$
means double in initial SOI value, and vice versa. As mentioned earlier in
Sec.1, SOI can be indicated in two forms that Rashba SOI can be controlled by
tuning a perpendicular electric field; so it will be reasonable to examine the
general foundation of the effect of SOI change, in order to study of materials
that show the same behavior as our example exhibited here.

|
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|
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Figure 4: (color online). (a) DFT band structure for a monolayer of BSb in terms of the coefficients of the initial strength of SOI (the numbers represent multiples of initial SOI): 1.0 (solid, black), 2.0 (yellow, dashed), 4.0 (blue, dotted), 4.2 (red, dashed-dot-dashed), 8.0 (magenta, dot-dot-dashed). (b), (c) Calculated LCNs in discretized Brillouin zone. Violet (dark) circles and blue (light) circles indicate LCNs of +1 and -1, respectively. The $\mathbb{Z}_{2}$ topological invariant is computed as the total LCNs modulo two in the half Brillouin zone [18], $\mathbb{Z}_{2}$ invariant is 0 and 1 for SOI=1 and SOI=4.2, respectively.  Table 2:  $\mathbb{Z}_{2}$ invariant and band-gap values by DFT for monolayer of BSb in terms of the coefficients of the initial strength of SOI (the numbers represent multiples of initial SOI). $\frac{SOI}{initial~{}SOI}$ | 1.0 | 2.0 | 4.0 | 4.2 | 8.0
---|---|---|---|---|---
$\epsilon_{g}(eV)$ | 0.36 | 0.29 | 0.12 | 0.10 | 0.00
$\mathbb{Z}_{2}$ invariant | 0 | 0 | 0 | 1 | 1

Next, as the Fig. 5 shows for monolayer of SiGe, changing the EF strength
perpendicular to the lattice plane, along the normal vector causes to
disappear the Dirac cone at a distinctive EF. $\mathbb{Z}_{2}$ invariant
calculation, as the total calculated LCNs modulo two in the half Brillouin
zone, indicates that at nearly $2.7~{}\frac{V}{\text{\AA}{}}$ of EF, SiGe
monolayer enters the trivial phase insulator, and the band-gap open
dramatically as reported in Table 3.

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Figure 5: (color online). Band structures of the monolayer of SiGe by tunning
EF strength, 0.0, 0.9 and 2.69 ($\frac{V}{\text{\AA}{}}$).

The intensity of the electric field used in these calculations is significant
because its changes indicate a topological phase transition, as an example is
reported by Sawahata et al. [18]. Clearly, this pattern is reproducible for
other materials with logical values of electric field strength.

Table 3:  band-gap $(eV)$ Vs. EF strength ($\frac{V}{\text{\AA}{}}$) for SiGe monolayer. $EF$ | ($\frac{V}{\text{\AA}{}}$) | -2.5 | -1.2 | -0.9 | -0.6 | -0.3 | 0.0 | 0.1 | 0.3 | 0.6 | 0.9 | 1.2 | 2.7
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\epsilon_{g}$ | DFT | 0.30 | 0.13 | 0.09 | 0.05 | 0.01 | 0.01 | 0.02 | 0.04 | 0.08 | 0.11 | 0.15 | 0.35
($eV$) | TBA | 0.08 | 0.06 | 0.06 | 0.06 | 0.06 | 0.07 | 0.07 | 0.08 | 0.09 | 0.11 | 0.14 | 0.35

##  6  Tight-binding approximation

In this section, with fitting the TB model and the DFT results, the Slater-
Koster parameters for 15, 2D honeycomb compounds of group-IV and group III-V
are calculated. Then, by considering the spin-orbit interaction and applying
the electric field, topological features, concluded in the previous section,
reproduce.

###  6.1  Tight binding parameterization from ab-initio calculations

Based on the review will be denoted in A, we have used the $sp^{3}$ TBA to
conclude the band structures. The result is illustrated in Fig. 6 for group
III-V compounds. As the plots show there is a good agreement with ab-initio
calculation. The used parameters have been summarized in Table 4, and the
results for group-IV compounds have been shown at Fig. 7, and the used
parameters have been summarized in Table 5.

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Figure 6: (color online). DFT Vs. TBA band structure for monolayer of (a) BN,
(b) BP, (c) BAs, (d) BSb, (e) BBi; DFT bands plotted in light, circle (blue);
and TBA bands plotted in dark, rectangle (red) points.
---
Figure 7: (color online). DFT Vs. TBA band structure for (a) graphene, (b)
silicene, (c) germanene, (d) stanene, and monolayer of (e) SiC, (f) SiGe, (g)
GeC, (h) SnC; DFT plotted in light, circle (blue); and TBA plotted in dark,
rectangle (red) points.

When a structure has a preference towards LB than PL, $sp^{2}$ hybridization
changes to $sp^{3}$. In PL mode the lobes of each atom are perpendicular to
the plane of the slab and this leads to the information of $\pi$ bonds with
nearest neighbors leading to conducting nature of the slab, but in LB
structures the lobes of neighboring atoms point in opposite directions so the
$\pi$ bands can only be formed with the second nearest neighbor than the first
neighbor. The $sp^{2}$ hybridized orbitals get slightly hybridized into
$sp^{3}$-like orbitals which causes weakening of $\pi$ bonds leading to
buckled structure of materials like silicene, germanene, stanene, and the
monolayer of SiGe, SnSi and SnGe; and BBi.

In this paper, for the purpose of simplicity and more efficiency, we have just
tried to derive the DFT bands considering only with first nearest neighbors.
Table 4, and Table 5 have been summarized all the parameters to obtain the
band structures by TBA for group III-V compounds, and group-IV combinations,
respectively. The fitting procedure is succeeding for group III-V compounds.
In the group-IV structures, some compounds ad-jointed tin such as monolayer of
SnC, SnSi and SnGe have been confronted with the inefficiency of the TB method
to the extent of the first nearest neighbor.

Table 4:  Slater-Koster parameters of BX honeycomb lattice derived by TBA. | BN | BP | BAs | BSb | BBi
---|---|---|---|---|---
$\epsilon_{s_{A}}$ | 0.359703 | -3.153820 | -3.315744 | -4.114327 | -3.418540
$\epsilon_{P_{y_{A}}}$ | 6.660174 | 2.849500 | 2.609751 | 2.062226 | -3.418540
$\epsilon_{P_{z_{A}}}$ | 2.116069 | 0.514927 | 0.435839 | 0.304630 | 0.187126
$\epsilon_{P_{x_{A}}}$ | 2.147729 | 0.448072 | 0.123560 | -0.120188 | 1.439313
$\epsilon_{s_{B}}$ | -10.465849 | -6.591012 | -7.395264 | -6.215011 | -8.434930
$\epsilon_{P_{y_{B}}}$ | 0.506664 | 1.166591 | 1.155927 | 1.406323 | 0.609970
$\epsilon_{P_{z_{B}}}$ | -2.329531 | -0.638155 | -0.213239 | -0.039482 | -0.067821
$\epsilon_{P_{x_{B}}}$ | 2.150085 | 1.281628 | 2.056722 | 1.955744 | 0.401624
$V_{ss\sigma}$ | -4.364859 | -3.124032 | -2.912592 | -2.497178 | -2.190991
$V_{sp\sigma}$ | 5.426549 | 3.405441 | 3.434569 | 2.923374 | 2.733426
$V_{pp\sigma}$ | 5.896137 | 3.821502 | 3.705766 | 3.215200 | 2.675127
$V_{pp\pi}$ | -2.341442 | -1.469154 | -1.548911 | -1.340630 | -1.256691
Table 5:  Slater-Koster parameters of the monolayer of IV-compounds derived by TBA. | Graphene | Silicene | Germanene | Stanene |
---|---|---|---|---|---
$\epsilon_{s_{A}}$ | -3.204545 | -4.342110 | -5.7160 | -4.819449 |
$\epsilon_{P_{y_{A}}}$ | 3.670414 | 1.900000 | 3.398440 | 2.477134 |
$\epsilon_{P_{z_{A}}}$ | 0.0 | 0.0 | 0.0 | 0.0 |
$\epsilon_{P_{x_{A}}}$ | 2.690589 | 1.451142 | 0.791040 | 1.443206 |
$\epsilon_{s_{B}}$ | - | - | - | - |
$\epsilon_{P_{y_{B}}}$ | - | - | - | - |
$\epsilon_{P_{z_{B}}}$ | - | - | - | - |
$\epsilon_{P_{x_{B}}}$ | - | - | - | - |
$V_{ss\sigma}$ | -5.485425 | -2.140001 | -1.755720 | -1.586277 |
$V_{sp\sigma}$ | 5.839044 | 2.454669 | 2.418080 | 2.115327 |
$V_{pp\sigma}$ | 6.479086 | 2.630116 | 2.555960 | 2.231675 |
$V_{pp\pi}$ | -2.749099 | -1.113355 | -0.86480 | -0.927208 |
| SiC | SiGe | GeC | SnC |
$\epsilon_{s_{A}}$ | -7.836687 | -5.313012 | -8.573873 | 0.602780 |
$\epsilon_{P_{y_{A}}}$ | -2.599611 | 2.837254 | -2.431318 | -2.052342 |
$\epsilon_{P_{z_{A}}}$ | 1.114450 | -0.029311 | 1.024802 | -10.278306 |
$\epsilon_{P_{x_{A}}}$ | -3.620749 | 1.614479 | -3.896555 | -0.969448 |
$\epsilon_{s_{B}}$ | -0.256636 | -4.500626 | -0.600946 | -0.643479 |
$\epsilon_{P_{y_{B}}}$ | 5.962847 | 0.932495 | 5.289562 | 19.998739 |
$\epsilon_{P_{z_{B}}}$ | -0.728725 | 0.0 | -0.644296 | 1.346508 |
$\epsilon_{P_{x_{B}}}$ | 3.524612 | 1.800000 | 3.511424 | -0.308166 |
$V_{ss\sigma}$ | -3.1421160 | -2.161956 | -3.068084 | -1.038965 |
$V_{sp\sigma}$ | -3.591319 | -2.651641 | -3.578150 | -1.754648 |
$V_{pp\sigma}$ | -2.1297328 | 2.655390 | -2.123841 | 3.018126 |
$V_{pp\pi}$ | 1.870858 | -1.077112 | 1.824391 | -2.345869 |

As well as reported the Fermi velocity by DFT calculation at Table 1, we focus
on the SiGe monolayer and by using this value will try to explore some
physical aspects by tight-binding model. Obviously, other well-worked fitted
results will be a topic for the continuation of this article.

Based on the TBA mentioned at A, to understand the origin of the band-gaps in
group III-V and group-IV materials, we can brief the TBA model considering the
hopping matrix elements for interaction between the $p_{z}$ orbitals on
nearest neighbor sites and neglecting any orbital overlaps. The Hamiltonian is

$H=\left(\begin{array}[]{cc}\epsilon_{p_{z}A}&t\cdot g(k)\\\ t\cdot
g(k)^{*}&\epsilon_{p_{z}B}\end{array}\right),$ (1)

where $p_{z}A$ and $p_{z}B$ are the onsite $p_{z}$ orbital energies of the A
and B atoms, respectively. And $t$ is the hopping integral as
$\frac{(\Delta^{2}V_{pp\sigma}+V_{pp\pi})}{\sqrt{1+\Delta^{2}}}$, and $g(k)$
defined as

$g(k)=e^{ik_{z}\Delta
a}(e^{-ik_{x}a}+2cos(\frac{\sqrt{3}k_{y}a}{2})e^{i\frac{k_{x}a}{2}}),$ (2)

here $k_{x}$ and $k_{y}$ are the wavevector components. The secular equation,
$det|H-E|=0$ , are solved for eigenvalues as

$E(k)=\frac{\epsilon_{p_{z}A}+\epsilon_{p_{z}B}}{2}\pm\left(\left(\frac{\epsilon_{p_{z}A}-\epsilon_{p_{z}A}}{2}\right)^{2}+(t\cdot
g(k))^{2}\right)^{1/2}.$ (3)

At the Dirac point, $K$ hight symmetry point, neglecting buckled parameter
compared to lattice constant, $g(k)^{2}=0$, so the energy band-gap is

$E_{g}=\left|\epsilon_{p_{z}A}-\epsilon_{p_{z}B}\right|.$ (4)

This equation reveals that the energy band-gap depends on the difference of
the energy between the $p_{z}$ orbitals of A and B. It is obvious that for
materials with similar A and B atoms, such as graphene, we encounter zero
band-gap.

###  6.2  Spin-orbit interaction and electric field in tight-binding model

The simple and effective $sp^{3}$ tight-binding model will be presented in
Sec. A in detail. The energy band structures have been gained, based on the
Slater-Koster parameters shown at the Table 4 and 5 for monolayers of group
III-V and IV materials, respectively, we convinced that the results, means the
band structure and band-gap, are in good agreement with the DFT results
displaced in the Sec. 5. In this section, we decide to peruse the topological
aspects of these materials in the presence of tunning parameters like SOI and
EF. So, with reference to this issue that topological feature of the band
structure of insulators is considered around the Fermi level, we would like to
examine the previous results, first in the presence of SOI, and second in the
presence of EF.

Table 6:  The band-gap (eV) values Vs. coefficients of initial SOI strength for a monolayer of BSb. For instance, $\times 2$ means double in initial SOI value, and vice versa. $\frac{SOI}{initial~{}SOI}$ |  | 1.0 | 2.0 | 4.0 | 4.2 | 8.0
---|---|---|---|---|---|---
$\epsilon_{g}$ | DFT | 0.36 | 0.29 | 0.12 | 0.10 | 0.00
$(eV)$ | TBA | 0.33 | 0.30 | 0.18 | 0.17 | 0.09
$\lambda_{SO_{B}}$ |  | 0.05 | $\times 2$ | $\times 4$ | $\times 4.2$ | $\times 8$
$\lambda_{SO_{Sb}}$ |  | 0.50 | $\times 2$ | $\times 4$ | $\times 4.2$ | $\times 8$

First, the band-gaps have been derived by TBA for a monolayer of BSb when the
intrinsic SOI strength is changed, besides the similar DFT results illustrated
in Table 6. It is obvious that the band-gap values descend as the SOI strength
increases. Here, the main purpose is to obtain the same results for band
structure with the TBA model. As Fig. 8 shows, this is carried out by
considering the Slater-Koster parameters, presented in Table 4, and changing
the SOI coefficients of Boron and antimony, presented as $\lambda_{so}$ in
Table 6. Therefore, the output for DFT and TBA has identical behavior that is
concluded our model is good enough to describe the band structure and
topological effects in the presence of SOI. This paves the way for further
investigation by TBA to provide an effective Hamiltonian near the Fermi level
around the $K$ high symmetry point.

---
Figure 8: (color online). TBA band structure for monolayer of BSb for
coefficients of SOI strength: 1.0 (solid, black), 2.0 (yellow, dashed), 4.0
(blue, dotted) , 4.2 (red, dashed-dot-dashed) , 8.0 (magenta, dot-dot-dashed).
---
Figure 9: (color online). The monolayer of SiGe band-gap with TBA and DFT Vs.
EF shows by square, red and diamond, black, respectively.

Second, the comparison between DFT and TBA about band-gap for SiGe monolayer
in the presence of EF announced by Fig. 10, shows the band-gap increases by
raising the EF strength. These values were presented in Table 3. At a border
value of EF around $2.7~{}\frac{V}{\text{\AA}{}}$ is larger than $0.3~{}eV$,
and the material enters the trivial phase. Our $sp^{3}$ tight-binding model
Hamiltonian needs to expand overlap integrals in the presence of EF, so we
need to use the Gaunt coefficients parameters [50, 51] to get the band
structure. The first and most important terms after applying EF, are on-site
contributions, named $\gamma_{sp}$, which is more discussed by C. R. Ast [27].
For each atoms, we have different $\gamma_{sp}$,

$\begin{array}[]{ll}\gamma_{sp_{Si}}=-a_{0}\frac{(2n^{*}_{Si}+1)n^{*}_{Si}}{2\sqrt{3}(Z_{Si}-s_{Si})}~{}=-0.513103~{}\text{\AA}{},\\\
\gamma_{sp_{Ge}}=-a_{0}\frac{(2n^{*}_{Ge}+1)n^{*}_{Ge}}{2\sqrt{3}(Z_{Ge}-s_{Ge})}=-0.840035~{}\text{\AA}{},\\\
\end{array}$ (5)

which in these relation $n^{*}$ is effective principle quantum number, 3 for
Si and 3.7 for Ge. Z is supposed to be the actual charge in the nucleus, and s
is a screening constant [22], 7.75 for Si and 26.35 for Ge. Moreover, other
overlap integrals in the presence of EF, $(\gamma_{sp2}-\gamma_{sp1})$ and
$\gamma_{pp1}$ calculated as 0.353372 Å and 0.141243 Å, respectively. The
appropriate strengths of intrinsic SOI coefficients are
$\lambda_{SO_{Si}}=0.014$ and $\lambda_{SO_{Ge}}=0.074$.

---
Figure 10: (color online). TBA for the monolayer of SiGe for EF
strength($\frac{V}{\text{\AA}{}}$) 0.0 (line, black), 0.9 (dashed, red), 2.7
(dotted, blue).

Because of using the GGA functional in DFT calculation and underestimating
band-gap size, an expectation of an exact matching between DFT and TBA result
is far from reality, whether for a monolayer of BSb or SiGe, but the general
behavior of both calculations is the same. As Fig. 9 shows regardless of the
magnitude of the band-gap, the behavior of its changes is consistent with the
change in the intensity of the EF between DFT and TBA. So. it is obvious that
the TBA model presented in the presence of EF has succeeded too.

##  7 Summary

In summary, based on the first-principles density functional theory, we have
investigated atomic structures, electronic structures, and topological
features of 2D binary materials, which composed of group III-V and IV-IV
elements, with planar and low-buckled geometries like graphene ($sp^{2}$) and
silicene ($sp^{3}$), respectively. Using the OpenMX package, which enabled us
to manipulate spin-orbit interaction and apply a perpendicular electric field,
we have explored the band-gap size, the existence of the Dirac point and the
topological phase transition, which are affected by these parameters, in some
compounds. In the case of the monolayer of BSb and SiGe, the band-gap closed,
and these materials transitioned to topological and trivial insulator phases,
respectively. As well as, the calculating $\mathbb{Z}_{2}$ invariant proved
these phase transitions. The other main result, based on a $sp^{3}$ tight-
binding model containing SOI and EF, is the set of Slater–Koster parameters,
which have been obtained from a fit to DFT calculations. These accurately have
reproduced the band structures and the topological patterns in the presence of
spin-orbit interaction and external electric field. In this way, our results
prepare a context for studying effective Hamiltonian and spintronic
applications of 2D topological nanostructures and can be quite promising for
nano-electronics.

##  8 Acknowledgments

We would like to thank Dr. S. Mahdi Fazeli (Department of Physics, University
of Qom, Qom, Iran) for his detailed scientific discussions and guidance in
this article. We would like also to show our gratitude to Dr. Hyun-Jung Kim
(Peter Grünberg Institut (PGI-1), Forschungszentrum Jülich, Germany) for
helpful discussions. We also thank Prof. I.I. Guseinov (Çanakkale Onsekiz Mart
Üniversitesi, Çanakkale, Turkey) for providing the initial program code of the
overlap integral over Slater-type orbitals.

##  Appendix A The tight-binding considerations

In this section, we will briefly introduce the tight-binding model has been
used for 2D honeycomb lattice, planar like as graphene, and buckled structures
like as silicene. As the Fig. 1 shows, the atom of type A has three first
nearest neighbor ($n.n.$) of the type B, and we neglect more further
neighbors. The basis sets of two-atoms unit cell contain $A_{0}$ and $B_{1}$
are: $\vec{a}_{1}=b(\frac{3}{2},\frac{\sqrt{3}}{2},\Delta\times\frac{a}{b})$
and $\vec{a}_{2}=b(\frac{3}{2},-\frac{\sqrt{3}}{2},\Delta\times\frac{a}{b})$,
which $a$ is the lattice constant and set to unity, and
$b=\frac{1}{\sqrt{3}}a$ is the distance of first $n.n.$. So if we consider the
atom of type A at the origin ($A_{0}:(0,0,0)$), three first $n.n.$ (B atoms)
located at $B_{1}:(-1,0,\Delta)b$ ,
$B_{2}:(\frac{1}{2},\frac{\sqrt{3}}{2},\Delta)b$ and
$B_{3}:(\frac{1}{2},\frac{-\sqrt{3}}{2},\Delta)b$. Also, we consider the
buckling parameter, $\Delta$, is lower than lattice constant, $\Delta<<a=1$,
and the structure is low-buckled. By using the Slater-Koster formalism [23];
considering $s$ and $p$ orbitals of A and B atoms; and overlap between them;
the $8\times 8$ Hamiltonian in the presence of an EF perpendicular to the
plane of lattice, will be as $H_{TB}$, where

$\begin{array}[]{lll}G_{0}=e^{-ik_{x}}e^{ik_{z}\Delta},\\\
G_{1}=2\cos\left(\frac{\sqrt{3}k_{y}}{2}\right)e^{i\frac{k_{x}}{2}}e^{ik_{z}\Delta},\\\
G_{2}=2i\sin\left(\frac{\sqrt{3}k_{y}}{2}\right)e^{i\frac{k_{x}}{2}}e^{ik_{z}\Delta},\end{array}$
(6)

are geometry functions, and

$\begin{array}[]{llll}A=\left(V_{pp\sigma}+3\Delta^{2}V_{pp\pi}\right),\\\
B=\left(V_{pp\sigma}-V_{pp\pi}\right),\\\
C=\left(V_{pp\sigma}+(3+4\Delta^{2})V_{pp\pi}\right),\\\
D=\left(3V_{pp\sigma}+(1+4\Delta^{2})V_{pp\pi}\right),\\\
F=\left(\Delta^{2}V_{pp\sigma}+V_{pp\pi}\right),\\\
R=\sqrt{1+\Delta^{2}},\end{array}$ (7)

A, B, C, D, F and R are the parameters.

$H_{TB}=\left(\begin{array}[]{rccccccccc}&S_{A}&P_{x_{A}}&P_{y_{A}}&P_{z_{A}}&S_{B}&P_{x_{B}}&P_{y_{B}}&P_{z_{B}}\\\
\\\ \hline\cr\\\ \\\
S_{A}\vline&\epsilon_{s_{A}}&0&0&-a_{0}E_{f}\frac{(2n_{A}+1)n_{A}}{2\sqrt{3}(Z_{A}-s_{A})}&V_{ss\sigma}\left(G_{0}+G_{1}\right)&\frac{V_{sp\sigma}\left(-G_{0}+\frac{1}{2}G_{1}\right)}{R}&\frac{V_{sp\sigma}\left(\frac{\sqrt{3}}{2}G_{2}\right)}{R}&\frac{\Delta
V_{sp\sigma}\left(G_{0}+G_{1}\right)}{R}\\\ \vline\\\
P_{x_{A}}\vline&0&\epsilon_{p_{x_{A}}}&0&0&\frac{V_{sp\sigma}\left(G_{0}-\frac{1}{2}G_{1}\right)}{R}&\frac{A\left(G_{0}\right)+\frac{C}{4}G_{1}}{R^{2}}&\frac{\frac{\sqrt{3}}{4}BG_{2}}{R^{2}}&\frac{\Delta
B\left(-G_{0}+\frac{1}{2}G_{1}\right)}{R^{2}}\\\ \vline\\\
P_{y_{A}}\vline&0&0&\epsilon_{p_{y_{A}}}&0&-\frac{V_{sp\sigma}\left(\frac{\sqrt{3}}{2}G_{2}\right)}{R}&\frac{\frac{\sqrt{3}}{4}BG_{2}}{R^{2}}&V_{pp\pi}\left(G_{0}\right)+\frac{DG_{1}}{4R^{2}}&\frac{\frac{\sqrt{3}}{2}\Delta
BG_{2}}{R^{2}}\\\ \vline\\\
P_{z_{A}}\vline&-a_{0}E_{f}\frac{(2n_{A}+1)n_{A}}{2\sqrt{3}(Z_{A}-s_{A})}&0&0&\epsilon_{p_{z_{A}}}&-\frac{\Delta
V_{sp\sigma}\left(G_{0}+G_{1}\right)}{R}&\frac{\Delta
B\left(-G_{0}+\frac{1}{2}G_{1}\right)}{R^{2}}&\frac{\frac{\sqrt{3}}{2}\Delta
DG_{2}}{R^{2}}&\frac{F(G_{0}+G_{1})}{R^{2}}\\\ \vline\\\
S_{B}\vline&V_{ss\sigma}\left(G^{\dagger}_{0}+G^{\dagger}_{1}\right)&\frac{V_{sp\sigma}\left(G^{\dagger}_{0}-\frac{1}{2}G^{\dagger}_{1}\right)}{R}&-\frac{V_{sp\sigma}\left(\frac{\sqrt{3}}{2}G^{\dagger}_{2}\right)}{R}&-\frac{\Delta
V_{sp\sigma}\left(G^{\dagger}_{0}+G^{\dagger}_{1}\right)}{R}&\epsilon_{s_{B}}&0&0&-a_{0}E_{f}\frac{(2n_{B}+1)n_{B}}{2\sqrt{3}(Z_{B}-s_{B})}\\\
\vline\\\
P_{x_{B}}\vline&\frac{V_{sp\sigma}\left(-G^{\dagger}_{0}+\frac{1}{2}G^{\dagger}_{1}\right)}{R}&\frac{A\left(G^{\dagger}_{0}\right)+\frac{C}{4}G^{\dagger}_{1}}{R^{2}}&\frac{\frac{\sqrt{3}}{4}BG^{\dagger}_{2}}{R^{2}}&\frac{\Delta
B\left(-G^{\dagger}_{0}+\frac{1}{2}G^{\dagger}_{1}\right)}{R^{2}}&0&\epsilon_{p_{x_{B}}}&0&0\\\
\vline\\\
P_{y_{B}}\vline&\frac{V_{sp\sigma}\left(\frac{\sqrt{3}}{2}G^{\dagger}_{2}\right)}{R}&\frac{\frac{\sqrt{3}}{4}BG^{\dagger}_{2}}{R^{2}}&V_{pp\pi}\left(G^{\dagger}_{0}\right)+\frac{DG^{\dagger}_{1}}{4R^{2}}&\frac{\frac{\sqrt{3}}{2}\Delta
DG^{\dagger}_{2}}{R^{2}}&0&0&\epsilon_{p_{y_{B}}}&0\\\ \vline\\\
P_{z_{B}}\vline&\frac{\Delta
V_{sp\sigma}\left(G^{\dagger}_{0}+G^{\dagger}_{1}\right)}{R}&\frac{\Delta
B\left(-G_{0}+\frac{1}{2}G^{\dagger}_{1}\right)}{R^{2}}&\frac{\frac{\sqrt{3}}{2}\Delta
BG^{\dagger}_{2}}{R^{2}}&\frac{F(G^{\dagger}_{0}+G^{\dagger}_{1})}{R^{2}}&-a_{0}E_{f}\frac{(2n_{B}+1)n_{B}}{2\sqrt{3}(Z_{B}-s_{B})}&0&0&\epsilon_{p_{z_{B}}}\end{array}\right).$
(8)

The perpendicular EF effect consider with $\lambda_{E_{f}}$ as strength;
$n_{A}(n_{B})$ and $Z_{A}(Z_{B})$ are effective quantum number and atomic
number of atom A(B), respectively; and $a_{0}$ is Bohr radius. The on-site
terms calculated as $\gamma_{sp}=<s|E_{f}z|p_{z}>$ by considering $z$ as
$rcos(\theta)$. Also, for orbital functions, we used the Slater spherical form
of them; and $n$ as effective quantum number; and $s$ as screening constant
[22]. Then other off-site terms related to overlap between neighbor atoms
consider as $H_{z,off-site}$, with $l,m,n$ as directional cosines depend on
the connecting vector.

$\begin{array}[]{llll}\gamma_{sp1}=<s|z|p_{z},b\hat{x}>,\\\
\gamma_{sp2}=<s,-\frac{\Delta}{2}\hat{z}|z|p_{z},\frac{\Delta}{2}\hat{z}>,\\\
\gamma_{pp1}=<p_{x}|z|p_{z},b\hat{x}>,\end{array}$ (9)

and

$H_{z,off-site}=$ | $E_{f}\left(\begin{tabular}[]{c}0\\\ $\frac{(l_{1}n_{1}G_{0}+l_{2}n_{2}G_{2})\left(\gamma_{sp2}-\gamma_{sp1}\right)}{R^{2}}$\\\ $\frac{(m_{1}n_{1}G_{0}+m_{2}n_{2}G_{2})\left(\gamma_{sp2}-\gamma_{sp1}\right)}{R^{2}}$\\\ $\frac{(\left(1-n_{1}^{2}\right)\gamma_{sp1}+n_{1}^{2}\gamma_{sp2})(G_{0}+G_{1})}{R^{2}}$\\\ \end{tabular}\right.$ |  | $\frac{(l_{1}n_{1}G_{0}+l_{2}n_{2}G_{2})\left(\gamma_{sp2}-\gamma_{sp1}\right)}{R^{2}}$
---
0
0
$-\frac{(l_{1}G_{0}+l_{2}G_{1})\gamma_{pp1}}{R^{2}}$
|  | $\frac{(m_{1}n_{1}G_{0}+m_{2}n_{2}G_{2})\left(\gamma_{sp2}-\gamma_{sp1}\right)}{R^{2}}$
---
0
0
$-\frac{(m_{1}G_{0}+m_{2}G_{2})\gamma_{pp1}}{R^{2}}$
$\left.\begin{tabular}[]{c}$\frac{(\left(1-n_{1}^{2}\right)\gamma_{sp1}+n_{1}^{2}\gamma_{sp2})(G_{0}+G_{1})}{R^{2}}$\\\
$\frac{(l_{1}G_{0}+l_{2}G_{1})\gamma_{pp1}}{R^{2}}$\\\
$\frac{(m_{1}G_{0}+m_{2}G_{2})\gamma_{pp1}}{R^{2}}$\\\ 0\\\
\end{tabular}\right)$,
(10)

more discussion prepared by C. R. Ast [27].

The SOI in the tight-binding method is traditionally taken into account only
for $p$ orbitals on interatomic matrix elements of the Hamiltonian [24]. We
use it in the form of the below matrix:

$H_{SOI}=\lambda_{so}\left(\begin{array}[]{rcccccc}&P_{x_{A}}^{\uparrow}&P_{x_{A}}^{\downarrow}&P_{y_{A}}^{\uparrow}&P_{y_{A}}^{\downarrow}&P_{z_{A}}^{\uparrow}&P_{z_{A}}^{\downarrow}\\\
\hline\cr P_{x_{A}}^{\uparrow}\vline&0&0&-$i$&0&0&1\\\
P_{x_{A}}^{\downarrow}\vline&0&0&0&$i$&-1&0\\\
P_{y_{A}}^{\uparrow}\vline&$i$&0&0&0&0&-$i$\\\
P_{y_{A}}^{\downarrow}\vline&0&-$i$&0&0&-$i$&0\\\
P_{z_{A}}^{\uparrow}\vline&0&-1&0&$i$&0&0\\\
P_{z_{A}}^{\downarrow}\vline&1&0&$i$&0&0&0\\\ \end{array}\right).$ (11)

Finally the overall Hamiltonian matrix is $H=H_{TB}~{}(include~{}H_{z,off-
site})+H_{SOI}$. It is obvious that when the SOI consider, the Hamiltonian
size changes to double ($16\times 16$).

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