# Direct band gap in gallium sulfide nanostructures

M. Mosaferi  I. Abdolhosseini Sarsari  M. Alaei  Department of Physics,
Isfahan University of Technology, Isfahan, 84156-83111, Iran

###### Abstract

The monolayer Gallium sulfide (GaS) was demonstrated as a promising two-
dimensional semiconductor material with considerable band gaps. The present
work investigates the band gap modulation of GaS monolayer under biaxial or
uniaxial strain by using Density functional theory calculation. We found that
GaS monolayer shows an indirect band gap that limits its optical applications.
The results show that GaS monolayer has a sizable band gap. The uniaxial
strain shifts band gap from indirect to direct in Gallium monochalcogenides
(GaS). This behavior, allowing applications such as electroluminescent devices
and laser. The detailed reasons for the band gap modulation are also discussed
by analyzing the projected density of states (PDOS). It indicates that due to
the role of py orbital through uniaxial strain become more significant than
others near the Fermi level. The indirect to direct band gap transition happen
at $\varepsilon$=-10y$\%$. Moreover, by investigating the strain energy and
transverse response of structures under uniaxial strain, we show that the GaS
monolayer has the Poisson’s ratio of 0.23 and 0.24 in the zigzag (x) and
armchair (y) directions, respectively. Thus, we conclude that the isotropic
nature of mechanical properties under strain.

GaS monolayer, band gap transition, strain

##  I INTRODUCTION

The 2D materials hold great promise in the design of new nano-devices for
future applications, due to their fascinating electronic, mechanical, optical,
and thermal properties which are caused by their low dimensionality and
quantum confinement effect. The some of these novel properties not seen in
their bulk counterparts Gupta  _et al._ (2015). 2D nano-sheets with hexagonal
structure, such as graphene, hexagonal boron nitride (h-BN) and transition
metal dichalcogenides (TMDs) have attracted extensive research effort in
recent years. The utility of these materials, however, is limited by some
imperfections such as the lack of a band-gap in graphene and the relatively
low mobility in some TMDs Radisavljevic  _et al._ (2011). It has motivated
continuing work in search of more 2D materials that demonstrate their
properties that may lead to improving specific performance.

Recently, another class of 2D materials, metal chalcogenides, came into being
and attracted considerable attention. These layered materials generally
possess the chemical formula MX, where M= Group IIIA and X= Group IVA. Gallium
sulfide is an indirect band gap semiconductor. A primitive cell of GaS
contains four Gallium atoms and four sulfur atoms. Its structure is formed
from four layered of S-Ga-Ga-S type as illustrated in Fig. 1. The interlayers
dominantly experience the weak Van der Waals interaction, while within the
layer the Ga and X atoms form the strong covalent bond similar to graphene and
TMDs.

GaS nanosheet was successfully isolated by micromechanical cleavage technique
Late  _et al._ (2012a)Aono  _et al._ (1993). Monolayer gallium sulfide has
been used in photodetectors, photoelectric devices, electrical sensors, near-
blue light emitting devices Shen  _et al._ (2009)Hu  _et al._ (2013)Aono  _et
al._ (1993), and field effect transistors (FET). The mobility of 0.1
$cm^{2}/V.S$ have been measured by David et al Late  _et al._ (2012b) in
ultra-thin GaS bottom-gate transistors. Moreover, the piezoelectric
coefficients of GaS monolayer are of the same order of magnitude as discovered
two-dimensional (2D) piezoelectric materials. (The boron nitride (BN) and
$MoS_{2}$ monolayers Li and Li (2015)).

The indirect band gap of GaS monolayer presents an obstacle for optoelectronic
devices applications involving light harvesting, light emitting,
photodetectors and lasers. The thicker thinfilm with direct optical band gap
is desired. The strain plays an important role to engineer band gap of
semiconductors in the field of microelectronic. It would be very interesting
to investigate the effect of strain on the band structure of GaS. The effect
of strain on graphene and TMDs have been widely studiedTopsakal  _et al._
(2010)Choi  _et al._ (2010)Peelaers and Van de Walle (2012)Scalise  _et al._
(2012) Ghorbani-Asl  _et al._ (2013)Sun  _et al._ (2016)Wang  _et al._ (2015).
The band structure of these can be remarkably tailored by applying strain Ni
_et al._ (2008)Huang  _et al._ (2011). If the energy difference of the
indirect and direct band gap is small, then it may be possible to achieve
indirect-direct band gap transition. The GaS nanoribbons can be directly
obtained by cutting the GaS monolayer (Fig. 8). Depending on the direction of
termination, there exist two kinds of nanoribbons: armchair and zigzag. The
armchair GaS nanoribbons and zigzag GaS nanoribbons can be identified by the
number of dimer lines or zigzag chains across the ribbon width and are labeled
as $N_{a}$-AGaSNR and $N_{z}$-ZGaSNR, respectively.

Figure 1: A schematic showing the crystal structure of GaS monolayer with (a)
top view and (b) side view. The blue and yellow spheres represent the Ga and S
atomic species, repectively. $a_{0}$ and $d$ represent the lattice constant
and the thickness of the GaS monolayer as defined by the distance between the
top and bottom layer of sulfur atom. The x and y orientations correspond to
the zigzag and armchair direction, respectively.

In the present work, we have studied the electronic properties of GaS
monolayer under strain. The strain can easily perform by a lattice mismatch on
the substrate or mechanical loadingShi  _et al._ (2013). we employed two type
of deformations: uniaxial and biaxial. The uniaxial strain is parallel to x
and y directions in the range of $\varepsilon$=-10$\%$ to
$\varepsilon$=+10$\%$. The biaxial compressive strain can firstly increase the
band gap, up until 2.76 eV at $\varepsilon$=-4$\%$ and then reduce it, while
biaxial tensile strain reduces the band gap gradually. Both of them can not
change the indirect band gap to direct band gap. Monolayer GaS undergoes
indirect to direct band gap transition under uniaxial compressive strain along
y-direction. It can indicate that as biaxial strain increasing the
semiconductor to metal transition occurs by the energy crossover of one state
in conduction band from Fermi level. Also, we focused on the effect of edge
passivation on the electronic band structures. We demonstrated that the
electronic properties of edge zigzag and armchair GaS nanoribbons can be
largely tuned by edge passivation. Therefore, offers a powerful tool for
customizing to a particular application. The band gap of nanoribbons can be
also reduced or enlarged, depending on the direction of termination (zigzag or
armchair) and width of nanoribbons.

##  II METHOD

The calculations were performed using ab initio density functional theory
(DFT) in conjunction with projector augmented wave (PAW) potentials Blöchl
(1994) and Perdew-Burke Ernzerted (PBE) generalized gradient approximation
(GGA) to the electronic exchange and correlation Perdew (1996), as implemented
in the Quantum Espresso (QE) package Giannozzi  _et al._ (2009). The K-point
mesh in the lateral directions was always 9$\times$9$\times$1 while, the
8$\times$1$\times$1 grid K-points was used for armchair nanoribbons and
1$\times$12$\times$1 for zigzag nanoribbons. A kinetic energy cutoff of 500 eV
was used for the plane-wave expansion. Test calculation with the bigger number
of K-points and higher cutoff gave essentially the same results. Bader
analysis was used to calculate charge transfer between Ga and S under
different strain by Critic2 Henkelman  _et al._ (2006).
The layered GaS is modeled using a supercell method with 2D periodic boundary
conditions. A vacuum region of 20 Å along the c direction was used to separate
the layers system in order to avoid spurious interactions. The geometric
structures of the GaS monolayer are optimized by the conjugate-gradient
minimization scheme. All of the atoms in the unit-cell are fully relaxed until
the force on each atom is minimized. Our calculated lattice constant for GaS
monolayer is 3.64 Å, and the band gap of 2.45 eV in good agreement with other
theoretical calculation Zolyomi  _et al._ (2013)Huang  _et al._ (2015).

##  III RESULTS AND DISCUSSIONS

Here, we examine the band gap evolution under biaxial and uniaxial strain. In
spite of DFT band gap underestimation in semiconductor, the Si nanostructures
strain-dependency is similar compared to optical band gap predicted by the
quasiparticle and advance configuration interaction methods Peng  _et al._
(2006). Therefore, we can predict the general trend of the strain effect on
band structure. The present work is mainly focused on the strain effect on the
electronic band structure by the GGA-PBE. As expected not only the Ga-S bond
length but also the distance between S atoms vary with applied strain, thus,
the band gap can be significantly tuned by biaxial and uniaxial strain as
shown in Fig. 2 and Fig. 4. So the GaS monolayer with an indirect band gap can
tune into a direct band gap with a suitable uniaxial strain and experiences a
semiconductor to metal transition with biaxial tensile strain.

The equilibrium structure was exposed to mechanical strain via two strategies:
First, biaxial deformation, 2D isotropic deformation on the hexagonal unit
cell with simultaneous change of $a_{0}$ and $b_{0}$ lattice vector. Second,
uniaxial deformation of $a_{0}$ ($b_{0}$) along the zigzag (armchair)
direction x (y) in the rectangular unit cell. The strain along the x and y
direction are given by

$\displaystyle\epsilon_{x}=(a_{x}-a_{0})/a_{0}$ (1)

and

$\displaystyle\epsilon_{y}=(b_{y}-b_{0})/b_{0}$ (2)

respectively. Where $a_{x}$ and $b_{y}$ are deformed lattice vector along x
and y directions. The calculated lattice parameters $a_{0}$ and $b_{0}$ in the
unstrained structure of GaS monolayer are 3.64 Å and 2.35 Å, respectively.

###  III.1 Biaxial strain

Our DFT calculations found that the strain-free GaS monolayer is the
semiconductor with an indirect bandgap that, the valence band maximum (VBM)
locates along the $\Gamma$-X line while the conduction band minimum (CBM) lies
at $\Gamma$ as shown in Fig. 2. To better understand the strain-depend band
gap behavior, we calculated the effect of biaxial strain on the band structure
was presented in Fig. 2. Note that, irreducible Brillouin Zone of GaS under
biaxial strain remains unchanged, as the hexagonal symmetry is saved. With
tensile biaxial strain, the energy of the CBM at $\Gamma$ reduces rapidly. But
the reduction of energy of VBM as labeled $\Lambda$ is slower than up to
+6$\%$. When more strain is added, the energy of $\Lambda$ (at VBM) increases.
The result indicates that when biaxial tensile strain is applied the CBM and
VBM always locate at $\Gamma$ and $\Lambda$ point, which moves close to the
Fermi level with an increase of tensile strain and suggesting a stronger
indirect band gap.

Figure 2: First-principles band structure of monolayer GaS under biaxial
tensile and compressive strain. The negative value indicates the compressive
strain. The VBM and CBM are labeled by A and B. The dashed lines are guide for
an eye for energy shifts of states. The horizontal dashed lines indicate the
Fermi level. The fermi level is set to zero. Figure 3: The projected density
of states of GaS monolayer under (a) biaxial tensile strain and (b) biaxial
compressive strain. The panels a-c and b-d refer to the Ga and S atoms,
respectively. The vertical dashed lines indicate the Fermi level. The Fermi
level is aligned at zero. Figure 4: The band structure calculated under
tensile and compressive strain along (a) zigzag and (b) armchair directions.
The variation of states A, B and C with strain are marked out by dashed lines.

Fig. 5 shows variation of band gap as a function of biaxial strain. It can be
found that, the band gap energy of GaS monolayer decrease approximately
linearly with increasing tensile strain. As the more tensile strain applied up
to $\varepsilon$=15$\%$, the band gap vanishes and consequently, the monolayer
becomes metallic with band crossing the Fermi level at the $\Gamma$ point. On
the other hand, when a compressive strain of $\varepsilon$=-2$\%$ was applied
to the structure, the VBM remaining at $\Lambda$ while the CBM shifted to M.
When $\varepsilon$=-4$\%$, the CBM of monolayer switches from M to K, while
keeps the VBM unchanged. It is obvious that the band at A point gradually
moves upward, consequently, at $\varepsilon$=-6$\%$ VBM changes to $\Gamma$
point. At the same time, the B (CBM) point energy downward under compressive
strain. As the value of the compressive strain increases similar behavior
occurs with the VBM and CBM. But the energy shift of B point is more prominent
than A point. Moreover, except within $\varepsilon$=-2$\%$ and
$\varepsilon$=-4$\%$ of compressive strain, the band gap generally reduced as
the strain increase. (shown on the Fig. 5). It is clear that the indirect band
gap can not be transformed to be direct band gap by applying biaxial strain in
the range of $\varepsilon$=-10$\%$ to $\varepsilon$=10$\%$.

Figure 5: (a) Band gap variation of GaS monolayer versus biaxial compressive
strain and tensile strain in the range of $\varepsilon$=-10$\%$ to
$\varepsilon$=10$\%$. The various gaps are indicated by the location in the
irreducible Brillouin Zone were the VBM and CBM occur. (b) Variation of
$\Delta{E}$ between the strain-free and strained GaS monolayer as function of
biaxial strain

To understand the bandgap engineering via biaxial strain, the projected
density of states (PDOSs) were calculated for the GaS monolayer. It can be
found that from Fig. 3, the electronic states of GaS monolayer near the CBM
and VBM contributed mainly from $p_{z}$ and $s$ orbitals of the Gallium atoms
and the $p_{y}$ orbital of the Sulfur atoms. It is clear that the S $p_{y}$
orbital plays a very important role in the VBM. As the tensile strain applied,
the CBM and the VBM shift toward the Fermi level. At $\varepsilon$=4$\%$, the
S $p_{y}$ orbital shifts back and VBM is replaced by S $p_{z}$ orbital. On the
other hand, when a compressive strain was applied of $\varepsilon$=-2$\%$ and
$\varepsilon$=-4$\%$, the CBM and VBM of sulfur atoms are not contributed by
the same electric components as that of pristine one. As shown in Fig. 3 at
$\varepsilon$=-2$\%$ and $\varepsilon$=-4$\%$, the CBM and VBM of sulfur atoms
are mainly contributed by $p_{z}$ orbital. At the same time, the electronic
component of VBM of Ga is not changed and it is still contributed by Ga $s$
and Ga $p_{z}$ orbitals, while the electronic components of CBM of Ga are
changed and contributed by Ga $p_{z}$ and $p_{y}$ orbitals. The main changing
in the electronic contribution of CBM and VBM of sulfur atoms, leading to a
band gap enhancement for GaS monolayer at $\varepsilon$=-2$\%$ and
$\varepsilon$=-4$\%$. As the compression increases further, above
$\varepsilon$=-4$\%$ the S and Ga $p_{y}$ orbitals shift toward the Fermi
level and the band gap decreases again.

###  III.2 Uniaxial strain

We have also applied uniaxial strain on rectangular GaS monolayer. Under
uniaxial strain, the original crystal symmetry is broken, thus the K-points
become no longer equivalent as before. We define the $\varepsilon_{x}$ and
$\varepsilon_{y}$ along the zigzag and armchair directions. In the band
structure of the pristine GaS monolayer, the VBM location between $\Gamma$-X
and the CBM location at the $\Gamma$ point results in indirect bandgap (see
Fig. 4).

Under zigzag compressive strain, the VBM is retained between $\Gamma$-X, as
the value of the compressive strain increase, the energy of A point gradually
increases. Beyond $\varepsilon$=-6x$\%$, A has a highest energy and thus
represent the VBM. With an increase compression, the dispersion relation of
the CBM at $\Gamma$ gradually deform, therefore making the pristine CBM point
at $\Gamma$ move to another energy point between $\Gamma$-X, above
$\varepsilon$=-4x$\%$. And the indirect semiconductor character is retained.
As the zigzag tensile uniaxial strain applied, the band gap retain indirect.
With continuous increase of the strain, the band gap decreases. As can be
found, the CBM and VBM remain at $\Gamma$ and between $\Gamma$-X,
respectively.

Figure 6: The PDOS on GaS monolayer under uniaxial compressive strain along y
(armchair) direction. The panels a and b refer to Ga and S atoms,
respectively.

Under the tensile strain along the y-direction (armchair) keeping the VBM
unchanged, while the energy of the CBM at $\Gamma$, decreases rapidly. Thereby
strengthening the indirect band gap quickly.

As the uniaxial compressive strain is taken along y direction, the CBM always
locates at the $\Gamma$ point with experiences approximately 0.7 eV downward
shift up to $\varepsilon$=-10y$\%$. At the same time, as the more compression
was applied, the energy of a sub-VB (labelled B) raises rapidly. At
$\varepsilon$=-10y$\%$, the energy of B at $\Gamma$ point has a higher energy
than C and becomes the VBM. Therefore showing a direct band gap at $\Gamma$.

In summary, the biaxial strain can not change the indirect band gap to direct,
Nevertheless, has more influence on electronic band structure. But the
armchair compressive uniaxial strain can reduce the band gap and leading to
indirect-direct transition at $\varepsilon$=-10y$\%$ strain. For more details,
the PDOS were calculated as shown in Fig. 6. By analyzing the PDOS under
armchair strain found that the role of $p_{y}$ orbital gradually increase for
both Ga and S atoms. With the continuous increase of the strain at
$\varepsilon$=-10y$\%$ (where indirect-direct transition happens) the CBM and
VBM are mainly contributed by S $p_{y}$ $p_{z}$ Ga $s$ $p_{y}$ orbitals.
Therefore, the band gap transition occurs at $\Gamma$ point caused by the
increasing contribution of $p_{y}$ orbital.

##  IV Charge transfer

In this section, we examine the charge transfer between Ga and S atoms By
Bader analysis Henkelman  _et al._ (2006). The depletion of charge on Ga atom
is 0.82e in pristine GaS monolayer, whereas excess charge on S atoms that
proves the ionicity in Ga-S bonding. The charge transfer from S to Ga
increases with increasing compressive biaxial strain. Thus, we conclude that
the Ga-S bonds become less ionic and mix ionic-covalent in comparison with
equilibrium structure. As the large tensile strain applied, (above
$\varepsilon$=6$\%$) the charge of sulfur and Gallium come back to near
isolate atom because of the distance between S and Ga increases.

##  V Mechanical properties

The strain in the x or y-direction and the transverse strain in the y or
x-direction are shown in Fig. 7. for the GaS monolayer Poission’s ratio:

$\displaystyle\nu=d\epsilon_{transverse}/d\epsilon_{axial}$ (3)

are calculated by fitting the figure of transverse strain response. The
obtained results are 0.23 and 0.24 under zigzag and armchair uniaxial strain,
respectively. These two close value of Poission’s ratio indicate the isotropic
nature of the mechanical properties of the GaS monolayer. In another word, we
understand the mechanical properties are independent of direction. While
phosphorene has anisotropic nature of mechanical properties with 0.24 and 0.7
value for Poission’s ration in zigzag and armchair direction Peng  _et al._
(2014).

In the Fig. 7, we plotted strain energy for two directions (zigzag and
armchair) $E_{strain}$ as:

$\displaystyle E_{strian}=E_{T}(\varepsilon)-E_{T}(\varepsilon=0)$ (4)

where $E_{T}(\varepsilon)$ and $E_{T}(\varepsilon=0)$ are the total energies
of the GaS monolayer with strain and strain-free, respectively. It is clear
that the near degeneracy of strain energy in zigzag and armchair directions in
the range of $\varepsilon$=-10$\%$ to $\varepsilon$=10$\%$, demonstrate that,
for applying strain in either zigzag or armchair directions need equal energy.
In addition, confirm the isotropic nature of mechanical properties. In the
end, we calculated the energy difference ($\Delta{E}$) between strained and
strain-free GaS monolayer as the function of biaxial strain.

The $\Delta{E}$ increases monotonously (Fig.5) for both tensile and
compressive strains, shows that the system is in the range of the elastic
deformation. Therefore, after removing applied strain, the structure releases
back to the original structure which is important for sensor applications.

Figure 7: (a) The applied axial strains in the zigzag and armchair direction,
respectively. (b) The evolution of strain energy of GaS monolayer under
uniaxial strain along the x (black) and y (dashed red) directons Figure 8: Top
and side views of the bare-edge (a) armchair GaS nanoribbon with $N_{a}$=6 and
(b) zigzag GaS nanoribbon with $N_{z}$=8. The width $N_{a}$ and $N_{z}$ of
AGaSNR and ZGaSNR are depicted by black dashed line. The blue and yellow
spheres represent Ga and S atoms, respectively.

##  VI Nanoribbon

We have also investigated the electronic properties of bare and hydrogen
saturated zigzag/armchair GaS nanoribbons to find indirect-direct band gap
transition. First, we calculated the band structure of bare-edge armchair GaS
nanoribbon with a width ranging from $N_{a}=4$ to $N_{a}=10$. We found that
all armchair nanoribbons are indirect band gap semiconductor. with Eg values
of approximately 1.47 eV, independent of the width, which is smaller than the
monolayer band gap(2.45 eV) and can not be interpreted by the well-known
quantum confinement effect. The new flat energy level at both conduction and
valence band edges came into being via edge atoms of nanoribbons presence.
engineering the band structure, both edges are passivated with hydrogen atoms
(each edge Ga and S atoms are terminated by one H atom). Fig. 9 depicts the
band structure for passivated armchair GaS nanoribbons. It reveals that the
AGaSNR remain semiconductor after hydrogen saturation and keep semiconducting
independent with the ribbon width. After passivation, the band gap experiences
a big jump from 1.47eV to 2.48 eV for all nanoribbons, approximately. As the
width of the nanoribbons increase, the band gap decrease, slowly. At the same
time, the band at $\Gamma$ point moves downward (not shown here) and become
CBM in 10-AGaSNR. Consequently, the band structure undergoes an indirect to
direct transitions in the 10-AGaSNR passivated. In addition, the energy bands
around the Fermi level are rather flat for all the investigated nanoribbons
Which suggest a large effective mass and improvement in their transport
properties.

Also, we examine the zigzag GaS nanoribbons with a width ranging from
$N_{z}$=4 to $N_{z}$=10. The results indicated that, in contrast to armchair
GaS nanoribbons, the zigzag structure are metallic and their metallic behavior
is independent of ribbon width. As can be found that from Fig. 9, when
saturating the zigzag nanoribbons edge with hydrogen, all of the nanoribbons
retain metallic behavior, except 4-ZGaSNR that become semiconductor with a
small indirect band gap of 0.16 eV and direct band gap of 0.15 eV. It caused
by edge effects in 4-ZGaSNR that influences each other.

Figure 9: (a) The calculated band structure of H-saturated 10-AGaSNR (Left)
and the relaxed H-saturated structure of 10-AGaSNR (Right). (b) The calculated
band structure of H-saturated 4-ZGaSNR (Left) and the relaxed H-saturated
structure of 4-ZGaSNR (Right). The blue, yellow and red spheres represent Ga,
S and H atoms. The Fermi level is aligned at zero.

##  VII CONCLUSION

In conclusion, the first-principles DFT calculations introduce some special
circumstance in which GaS structures experience indirect-direct band gap
transition. Applying the mechanical deformation via biaxial and uniaxial
strain, not only the Ga-S bond length, but also the angle between S-Ga-S atoms
have a significant impact on the band structure of GaS monolayer. We show that
under biaxial strain in the range of $\varepsilon$=-10$\%$ to 10$\%$, the
indirect to direct transition of GaS monolayer did not observe. The band gap
always decreases under strain except at $\varepsilon$=-2$\%$ and
$\varepsilon$=-4$\%$ because the $p_{z}$ orbital shift towards the Fermi level
and above $\varepsilon$=-4$\%$, replaced by $p_{y}$ again and the GaS
monolayer exhibits a semiconductor to metal transition at
$\varepsilon$=+15$\%$. Under uniaxial compressive strain along y (armchair)
direction, the upward shift of sub-VBM is more than VBM and become VBM at
$\Gamma$ point and leading to crossover from indirect to direct band gap at
$\varepsilon$=-10$\%$ due to $p_{y}$ orbital move toward the Fermi level in
both Ga and S atoms.
The Poission’s ratio along armchair and zigzag directions has isotropic nature
and we can tailor GaS band gap with uniaxial $\varepsilon$=-10$\%$ strain
without concern to mechanical anisotropy in contrast to phosphorene that
posses anisotropic nature of mechanical properties.
For nanoribbon structures, we find that both bare and hydrogen saturated
AGaSNR are a semiconductor, while the ZGaSNRs are metallic. The semiconductor
behavior of the AGaSNR is independent of the ribbon width. The band gaps of
H-saturated AGaSNR increases in comparison to bare-edge and the size of band
gap do not vary significantly with ribbon width. The indirect to direct band
gap transition observed in 10-AGaSNR. when the armchair ribbon edges are
saturated with hydrogen atoms. The 4-ZGaSNR exhibits an indirect band gap of
0.16eV and the direct band gap of 0.15eV. Another H-saturated $N_{z}$-ZGaSNR
with $N_{z}$=6 - 8 - 10 retain metallic behavior. The improvement towards
controlling the band structure and optoelectronic properties of GaS monolayer
achieved with strain engineering.

##  VIII ACKNOWLEDGMENTS

We would like to acknowledge the Isfahan University of Technology. The authors
gratefully acknowledge the Sheikh Bahaei National High Performance Computing
Center (SBNHPCC) for providing computing facilities and time. SBNHPCC is
supported by scientific and technological department of presidential office
and Isfahan University of Technology (IUT).

## References

  * Gupta  _et al._ (2015) A. Gupta, T. Sakthivel, and S. Seal, Progress in Materials Science 73, 44 (2015).
  * Radisavljevic  _et al._ (2011) B. Radisavljevic, A. Radenovic, J. Brivio, i. V. Giacometti, and A. Kis, Nature nanotechnology 6, 147 (2011).
  * Late  _et al._ (2012a) D. J. Late, B. Liu, H. R. Matte, C. Rao, and V. P. Dravid, Advanced Functional Materials 22, 1894 (2012a).
  * Aono  _et al._ (1993) T. Aono, K. Kase, and A. Kinoshita, Journal of applied physics 74, 2818 (1993).
  * Shen  _et al._ (2009) G. Shen, D. Chen, P.-C. Chen, and C. Zhou, Acs Nano 3, 1115 (2009).
  * Hu  _et al._ (2013) P. Hu, L. Wang, M. Yoon, J. Zhang, W. Feng, X. Wang, Z. Wen, J. C. Idrobo, Y. Miyamoto, D. B. Geohegan, _et al._ , Nano letters 13, 1649 (2013).
  * Late  _et al._ (2012b) D. J. Late, B. Liu, J. Luo, A. Yan, H. R. Matte, M. Grayson, C. Rao, and V. P. Dravid, Advanced Materials 24, 3549 (2012b).
  * Li and Li (2015) W. Li and J. Li, Nano Research 8, 3796 (2015).
  * Topsakal  _et al._ (2010) M. Topsakal, S. Cahangirov, and S. Ciraci, Applied Physics Letters 96, 091912 (2010).
  * Choi  _et al._ (2010) S.-M. Choi, S.-H. Jhi, and Y.-W. Son, Physical Review B 81, 081407 (2010).
  * Peelaers and Van de Walle (2012) H. Peelaers and C. G. Van de Walle, Physical Review B 86, 241401 (2012).
  * Scalise  _et al._ (2012) E. Scalise, M. Houssa, G. Pourtois, V. Afanas’ev, and A. Stesmans, Nano Research 5, 43 (2012).
  * Ghorbani-Asl  _et al._ (2013) M. Ghorbani-Asl, S. Borini, A. Kuc, and T. Heine, Physical Review B 87, 235434 (2013).
  * Sun  _et al._ (2016) Y. Sun, D. Wang, and Z. Shuai, The Journal of Physical Chemistry C 120, 21866 (2016).
  * Wang  _et al._ (2015) Y. Wang, C. Cong, W. Yang, J. Shang, N. Peimyoo, Y. Chen, J. Kang, J. Wang, W. Huang, and T. Yu, Nano Research 8, 2562 (2015).
  * Ni  _et al._ (2008) Z. H. Ni, T. Yu, Y. H. Lu, Y. Y. Wang, Y. P. Feng, and Z. X. Shen, ACS nano 2, 2301 (2008).
  * Huang  _et al._ (2011) B. Huang, J. Yu, and S.-H. Wei, Physical Review B 84, 075415 (2011).
  * Shi  _et al._ (2013) H. Shi, H. Pan, Y.-W. Zhang, and B. I. Yakobson, Physical Review B 87, 155304 (2013).
  * Blöchl (1994) P. E. Blöchl, Physical review B 50, 17953 (1994).
  * Perdew (1996) J. P. Perdew, Phys. Rev. Lett. 77, 3865 (1996).
  * Giannozzi  _et al._ (2009) P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, _et al._ , Journal of physics: Condensed matter 21, 395502 (2009).
  * Henkelman  _et al._ (2006) G. Henkelman, A. Arnaldsson, and H. Jónsson, Computational Materials Science 36, 354 (2006).
  * Zolyomi  _et al._ (2013) V. Zolyomi, N. Drummond, and V. Fal’Ko, Physical Review B 87, 195403 (2013).
  * Huang  _et al._ (2015) L. Huang, Z. Chen, and J. Li, RSC Advances 5, 5788 (2015).
  * Peng  _et al._ (2006) X.-H. Peng, S. Ganti, A. Alizadeh, P. Sharma, S. Kumar, and S. Nayak, Physical Review B 74, 035339 (2006).
  * Peng  _et al._ (2014) X. Peng, Q. Wei, and A. Copple, Physical Review B 90, 085402 (2014).