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1. Effect of tensile strain on the band structure and carrier transport of germanium
monosulphide monolayer: a first-principles study
Guanxing Guo1,2, Gang Bi1 ✉
1
School of Information and Electrical Engineering, Zhejiang University City College, Hangzhou 310015, People’s Republic
of China
2
College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310027, People’s Republic
of China
✉ E-mail: bigang@zju.edu.cn
Published in Micro & Nano Letters; Received on 13th October 2017; Accepted on 15th January 2018
The electronic properties of germanium monosulphide (GeS) monolayer under tensile strain were investigated using first-principles
calculations. Our computations showed that the band gap of GeS monolayer was tuned from 1.96 to 2.72 eV via uniaxial and biaxial
tensile strain in the range of 10%. Besides, two transitions involving indirect to direct and direct to indirect were triggered when GeS
monolayer was applied 3.5 and 9% tensile strain in the zigzag direction, however this transition had not happened when the tensile strain
was applied in the armchair and biaxial direction. The band gap variations of GeS monolayer with the tensile strain were explained using
a bond nature mechanism based on the Heitler–London’s exchange energy model. Moreover, upon applying external strain, the acoustic
phonon limited carrier mobility of GeS monolayer had an enhancement with more than two orders of magnitude at 300 K, from
2.40 × 103
to 8.11 × 105
cm2
V−1
s−1
. These findings show that strain engineering is an effective way to tune the electronic properties of
GeS monolayer and to extend the applications of GeS monolayer in the field of electronics and optoelectronics.
1. Introduction: Two-dimensional (2D) layered materials with
strong in-plane chemical bonds and weak out-of-plane van der
Waals interactions have been extensively studied due to their wide
range of electronic, optical, mechanical, and electrochemical proper-
ties [1–4]. For example, graphene, the first 2D atomic crystal, is the
thinnest and strongest material and has an ultrahigh carrier mobility
exceeding 105
cm2
V−1
s−1
at room temperature [5, 6]. However,
graphene is a semimetal with zero band gap, which is not suitable
for electronic applications such as field-effect transistors (FETs).
Compared with graphene, MoS2 monolayer, a typical 2D material
of transition metal dichalcogenides, possesses a direct band gap
of 1.8 eV and a carrier mobility of 200 cm2
V−1
s−1
[7, 8], which
is more promising for electronic applications. Nevertheless,
the carrier mobility of MoS2 monolayer is still too low for high-
performance FET applications. Most recently, phosphorene, the
single- or few-layer form of black phosphorus, is explored with a
direct band gap from 0.3 to 2.0 eV and a high hole mobility up to
1000 cm2
V−1
s−1
and holds great promise for applications in elec-
tronics and optoelectronics [9–11]. Still, new 2D layered materials
with moderate band gap and high carrier mobility are desired for
electronics and optoelectronics applications.
Quite recently, monolayer group-IV monochalcogenides (e.g.
GeS, GeSe, SnS, and SnSe) with the isoelectronic counterpart of
black phosphorus have attracted wide research interest due to their
unique anisotropic properties [12–14]. Especially, germanium
monosulphide (GeS) is a narrow band gap semiconductor with the
band gap in the range of 1.55–1.65 eV [15]. Besides, GeS has
advantages of high chemical and environmental stability, low cost,
earth abundant and environmentally friendly elements, making it a
promising material for applications in photovoltaics [16], photode-
tectors [17], and lithium ion batteries [18]. In addition, previous
first-principles calculations show that GeS monolayer is an indirect
band gap semiconductor with kinetic and thermodynamic stability
and has an electron mobility up to 3680 cm2
V−1
s−1
, which is
higher than that of MoS2 monolayer and phosphorene, making it a
promising 2D material for applications in electronics [19].
On the other hand, modulating the electronic properties of semi-
conductor materials is important to extend their applications.
The extern strain is a common way to tune the electronic properties
of materials such as graphene [20], MoS2 [21], phosphorene [22],
and TiS3 [23]. Although the band gap of GeS monolayer is also pre-
dicted to be tuned via external strain [24], its mechanism is unclear
at present. Besides, whether the carrier mobility of GeS monolayer
can be enhanced by the extern strain has not been reported.
To address these questions, we studied the tensile strain effect on
the electronic properties of GeS monolayer based on the density
functional theory (DFT). The near-band-edge energy shifts and
corresponding band gap variations with the tensile strain were
explained using a bond nature mechanism based on the Heitler–
London’s exchange energy model. Moreover, an enhancement
of the acoustic phonon limited carrier mobility based on the de-
formation potential theory and the effective mass approximation
was found at the certain tensile strain. Our calculations show that
strain engineering can dramatically tune the electronic properties
of GeS monolayer and further broaden the applications of GeS
monolayer in electronics and optoelectronics.
2. Computational methods: First-principles calculations were
carried out using the DFT [25, 26] within the generalised
gradient approximation [27] of the Perdew–Burke–Ernzerhof
(PBE) [28] exchange correlation functional as implemented in the
Vienna ab initio simulation package [29, 30]. The DFT-D2 [31]
method of Grimme was adopted to account for the van der Waals
(vdW) interactions. The projector augmented wave method was
here used to deal with the electron-ion interaction [32], and the
plane wave energy cutoff was set to 500 eV. For bulk GeS, a
7 × 3 × 7 Monkhorst–Pack [33] grid was used for geometrical
optimisation while a 21 × 9 × 21 grid was used for electronic
structure calculations. As for GeS monolayer, 7 × 1 × 7 and
21 × 1 × 21 Monkhorst–Pack grids were used for geometrical
optimisation and electronic structure calculations, respectively. A
vacuum layer 20 Å was taken to avoid interaction between two
adjacent images. The structures were fully relaxed until the
Hellmann–Feynman force on each atom was 0.01 eV A–1
and
the total energy change was 1.0 × 10−5
eV. Since the PBE
functional tends to underestimate the band gap of
600
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Micro Nano Letters, 2018, Vol. 13, Iss. 5, pp. 600–605
doi: 10.1049/mnl.2017.0733

2. semiconductors, the Heyd–Scuseria–Ernzerhof (HSE06) [34]
hybrid functional was also used to get better band gap values.
The carrier mobility (m) was calculated using the deformation po-
tential theory and the effective mass approximation [35], which
have been applied to study the mobility of low-dimensional
systems [19, 36, 37]. The carrier mobility of two-dimensional
systems is given by
m =
eh
− 3
C
kBTm∗mdE2
d
, (1)
where e is the electron charge, h
− is the reduced Planck constant,
kB is the Boltzmann constant, and T is the temperature. C is
the elastic modulus along the transport direction defined as
C = (∂2
Etotal/∂2
d)/S0, in which Etotal is the total energy of the
system, d is the small strain applied in the transport direction, and
S0 is the area of the optimised system. Ed is the deformation poten-
tial constant defined as Ed = ∂Eedge/∂d, where Eedge is the energy
of the band edge position with respect to the core level under a
small strain d. m∗
is the carrier effective mass calculated from
m∗
= h
− 2
(∂2
E/∂2
k)−1
, in which E is the energy near the conduction
band minimum (CBM) or the valence band maximum (VBM), and
k is the magnitude of the wave-vector in momentum space. md is the
average effective mass calculated from md =
m∗
am∗
c
√
, in which m∗
a
and m∗
c are the effective mass in the a and c directions, respectively.
3. Results and discussion
3.1. Electronic properties of unstrained system: Bulk GeS (Fig. 1a)
has an orthorhombic structure belonging to the Pbnm space
group with the D16
2h point group. The bulk GeS has a unique
structural characteristic with a puckered configuration along the
a (armchair) direction and a bilayer structure along the c (zigzag)
direction. The DFT-D2 optimised lattice constants of bulk GeS
are a = 4.32, b = 10.66, and c = 3.68 Å, in good agreement with
the experiment values, a = 4.30, b = 10.48, and c = 3.65 Å [38].
Both PBE and HSE06 calculations indicate that the bulk GeS is a
direct band gap semiconductor at the G point of the first Brillouin
zone (Fig. 1b). The PBE band gap is 1.19 eV (Fig. 1c). It is well
known that the PBE functional underestimates the band gap
of semiconductors while the HSE06 functional gives a quite
good value. The HSE06 band gap is 1.82 eV, which is larger
than the PBE value and close to the measured optical direct gap
of 1.61 eV [16]. The good agreement between our calculations
and experiments for the bulk GeS shows that the theoretical
method used here is reliable.
Fig. 2 shows the PBE band structure of GeS monolayer. The
lattice constants of GeS monolayer are optimised to be a = 4.41
and c = 3.65 Å. The PBE and HSE06 band gaps are 1.60 and
2.28 eV, respectively, in good agreement with the previous calcula-
tions [19, 24]. Different from the bulk GeS, GeS monolayer is an
indirect band gap semiconductor whose VBM is located on the
G–X path whereas the CBM on the G–Z path. Notably, the PBE
direct band gaps of GeS monolayer at the G point and the VBM
are both 1.96 eV, quite close to that of the indirect band gap.
The carrier mobility and related parameters of GeS monolayer are
obviously anisotropic (Table 1), which are basically in agreement
with previous calculations [19]. The electron mobility is larger
than the hole mobility and has a high value of 2400 cm2
V−1
s−1
in the a direction. Besides, the elastic modulus in the a direction
is smaller than that in the c direction, indicating that the a direction
is softer than the c direction.
3.2. Tensile strain effect on the band structure and band gap: The
tensile strain was applied to the optimised GeS monolayer by
manually changing the lattice constant in the a or c direction. The
lattice constant in the transverse direction and the internal atomic
positions were fully relaxed as the response to the external strain.
Although the PBE functional is known to underestimate the band
gap of semiconductors, it has a similar band structure with the
HSE06 functional, which gives an improved prediction of the
band gap. Besides, it is proved that the PBE functional correctly
predicted the general trends of the strain effect on the band structure
and near-band-edge state in phosphorene [22]. Thus, the electronic
properties of GeS monolayer under tensile strain were only calcu-
lated using the PBE functional in this work for saving time.
Fig. 3 shows the effect of tensile strain on the band structure of
GeS monolayer. It is clear that GeS monolayer keeps the indirect
band gap under tensile strain along the a direction and biaxial
tensile strain. However, GeS monolayer transforms to the direct
Fig. 1 Illustration of crystalline polymorphs, high-symmetry points, and
band structure of bulk GeS
a 3 × 1 × 3 supercell structure of GeS, bigger grey and smaller yellow balls
represent Ge and S atoms, respectively
b First Brillouin zone and high-symmetry points of bulk GeS
c PBE band structure of bulk GeS
Fig. 2 PBE band structure of GeS monolayer
Table 1 Carrier effective mass m*, elastic modulus C, deformation
potential constant Ed, and carrier mobility m of GeS monolayer
Carrier
type
Direction m*(me) C (N m–1
) Ed (eV) m(×103
cm2
V−1
s−1
)
electron a 0.22 16.61 −1.46 2.40
c 0.45 52.50 2.00 1.97
hole a 0.26 16.61 −6.02 0.10
c 0.54 52.50 −9.66 0.06
Micro Nano Letters, 2018, Vol. 13, Iss. 5, pp. 600–605
doi: 10.1049/mnl.2017.0733
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3. band gap when the tensile strain along the c direction increases to
4% and returns to the indirect band gap at 9%.
There are some states (labelled as A–E in Fig. 3) competing for
the CBM and VBM in defining the band gap of GeS monolayer
under tensile strain. The energies of these near-band-edge states
with respect to the vacuum level under tensile strain are shown in
Fig. 4.
As shown in Fig. 4a, state B has a higher energy than state C
when the tensile strain along the a direction is less than the critical
value of 4% and represents the VBM. However, the energy of the
state C rises gradually when the strain continues to increase and
becomes the VBM. Since the VBM decrease faster than the CBM
with the increasing of tensile strain, the band gap of GeS monolayer
is expected to increase.
For the tensile strain applied in the c direction, state A and state D
compete for the CBM while state B and state E compete for the
VBM as shown in Fig. 4b. When the tensile strain is 3.5%, state
A has a lower energy than state D and represents the CBM.
Similarly, state B represents the VBM until the tensile strain
increases to 9%. With the increasing of tensile strain, the energy
of state E will exceed that of state B and becomes the VBM.
Thus, the band gap of GeS monolayer is predicted to decrease
with the increasing of tensile strain along the c direction.
As for the biaxial tensile strain, state A and state D compete for
the CBM while state B and state C compete for the VBM as shown
in Fig. 4c. The former two have a similar trend with the uniaxial
tensile strain along the c direction but a critical strain of 3%. The
latter two have a similar trend with the uniaxial tensile strain
along the a direction but a critical strain of 2% for state B decreases
dramatically than state C. Therefore, the band gap of GeS mono-
layer is expected to increase firstly and then decrease.
Fig. 5 shows the PBE band gap variations of GeS monolayer
under tensile strain. It can be seen that for the tensile strain
applied in the a direction, as the tensile strain increases, the band
gap of GeS monolayer increases. For the tensile strain applied in
the c direction, the band gap increases at the onset of tensile
strain and then decreases when the tensile strain exceeds 4%. As
for the biaxial tensile strain, the band gap of GeS monolayer
increases dramatically when the tensile strain is 3%, and then
decreases with the increasing of tensile strain. The PBE band gap
of GeS monolayer can be tuned from 1.37 to 1.94 eV and the
corresponding HSE06 band gap range is 1.96–2.72 eV.
3.3. Tensile strain effect on the near-band-edge orbitals: To
understand the trends of states A–E under tensile strain in Fig. 4,
their charge densities are plotted in Fig. 6. The different energy
shifts with strain are closely related to the bonding/antibonding
nature of the orbitals [22]. The energies of the bonding and
antibonding states in the Heitler–London’s model are given by
Ebonding = 2E0 +
e2
R
+
K + H
1 + S2
, (2)
Eantibonding = 2E0 +
e2
R
+
K − H
1 − S2
, (3)
Fig. 3 PBE band structures of GeS monolayer
a Under uniaxial tensile strain along the a direction
b Under uniaxial tensile strain along the c direction
c Under biaxial tensile strain
Fig. 4 Energies of the near-band-edge states A–E with respect to the vacuum level
a Under uniaxial tensile strain along the a direction
b Under uniaxial tensile strain along the c direction
c Under biaxial tensile strain
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doi: 10.1049/mnl.2017.0733

4. where E0 is the energy for an isolated atom, e is the electron charge,
R is the distance between two adjacent atoms, K is the classical
Coulomb energy between the electron–electron and electron–ion
interactions, and S is the overlap integral of the orbitals between
different atomic sites, which is usually much smaller than 1,
thus the square of S is even smaller. Therefore, the energies
of the bonding and antibonding states mainly depend on the
exchange integral term H. Also, the exchange H increases with
the increasing of external strain, resulting in the bonding energy
Ebonding to increase while the antibonding energy Eantibonding to
decrease shown in Fig. 6h. As for the non-bonding states, the
energy is expected to be insensitive to strain and is almost fixed
for the wave function overlap is minimal.
To further determine the bond nature of these states, Fig. 7a
shows the partial density of state (PDOS) of unstrained GeS mono-
layer, it can be seen that state A is dominated by pz orbital of Ge
mixed with s and px orbitals of S, state B is dominated by px
orbital of S mixed with s orbital of Ge. As shown in Fig. 7b,
state C is dominated by py orbital of S mixed with py and s orbitals
of Ge, state D is dominated by px orbital of Ge mixed with s orbital
of S. As seen from Fig. 7c, state E is dominated by pz orbital of S
mixed with s orbital of Ge.
Combined the charge densities of the near-band-edge states
(Fig. 6) with the PDOS of GeS monolayer (Fig. 7), it can be seen
that state A is antibonding in the a direction while bonding in
the c direction (Figs. 6a and b). State B is antibonding in the a dir-
ection and non-bonding in the c direction (Figs. 6c and d). State C
and state D are both antibonding in the a and c directions, respect-
ively (Figs. 6e and f ). State E is bonding in the c direction (Fig. 6g).
Therefore, bonding state A and state E in the c direction are
expected to increase with tensile strain. Antibonding states A, B,
C in the a direction and state D in the c direction are expected to
decrease with tensile strain. Non-bonding state B in the c direction
is almost unchanged with tensile strain. As for the biaxial tensile
strain, the energies of near-band-edge states A–D are a combination
of the uniaxial tensile strain. For example, the energy of state A
increases with the biaxial tensile strain for its increase in the c dir-
ection is faster than the decrease in the a direction. The analyses
above are consistent with the results in Fig. 4.
3.4. Tensile strain effect on the carrier effective mass and mobility:
The acoustic phonon limited carrier mobility at 300 K under tensile
strain was calculated using (1). The related parameter, carrier
effective mass, is presented in Fig. 8. For the tensile strain
along the a direction, the effective mass of the electron in the
a direction is almost unchanged while the effective mass of the
electron in the c direction increases slowly with the increasing of
the strain. The effective mass of the hole along the a direction
increases as the tensile strain increases to 4%, and then remains
unchanged. The effective mass of the hole in the c direction has
a dramatic increase at the same strain and then increases with
the strain. The trends of the effective mass of the hole when the
tensile strain is applied in the a direction are the consequence of
the strain effect on the band structure in Fig. 3, where the VBM
transforms from state B to state C at the critical strain of 4%.
The variation of the carrier effective mass gets more complicated
when the tensile strain is applied along the c direction. Under the
circumstances, four states in the CBM and VBM will raise
Fig. 5 PBE band gap of GeS monolayer under tensile strain along the a dir-
ection (black square) and c direction (red circle) and biaxial tensile strain
(green triangle)
Fig. 6 Charge densities of the near-band-edge states and schematic of
energy response in GeS monolayer
a–g States A–E, bigger grey and smaller yellow balls represent Ge and S
atoms, respectively
h Schematic of energy response to axial strain
Fig. 7 PDOS of GeS monolayer
a PDOS of unstrained GeS monolayer
b PDOS of GeS monolayer under biaxial tensile strain of 10%
c PDOS of GeS monolayer under uniaxial tensile strain along the c direction
of 10%
Micro Nano Letters, 2018, Vol. 13, Iss. 5, pp. 600–605
doi: 10.1049/mnl.2017.0733
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5. competition issues. The effective mass of the electron in the a dir-
ection is almost fixed, while that in the c direction decreases in the
range of 3% and then remains unchanged. The effective mass of
the hole in the a direction increases slowly and has a decrease
at the strain of 9%, while that in the c direction decreases distinctly
but has an increase at the same strain. The effective mass of the
electron in the c direction results from the crossing of state A and
state D at the tensile strain of 3.5%. The variation of the effective
mass of the hole at the tensile strain of 9% is the consequence of
the transformation of the VBM from state B to state E.
The variation of the carrier effective mass under biaxial tensile
strain is similar to that under uniaxial tensile strain along the a dir-
ection because the former has the same near-band-edge states
except for a new state D. Besides, the former has a different critical
strain of 3% for the effective mass of the hole.
The variation of the carrier mobility under tensile strain is more
complex than that of the carrier effective mass because the former
also depends on the elastic modulus and deformation potential con-
stant. There are some enhancements of the carrier mobility at some
certain tensile strains shown in Fig. 9. For example, when the tensile
strain is applied in the a direction, a peak value of the electron
mobility in the a direction is 1.13 × 104
cm2
V−1
s−1
at the strain
of 10%. For the tensile strain along the c direction, the electron
mobility in the a direction is 8.11 × 105
cm2
V−1
s−1
at the strain
of 6%. As for the biaxial tensile strain, the electron mobility in
the a direction is 4.13 × 103
cm2
V−1
s−1
at the strain of 10%.
4. Conclusions: Using first-principles calculations, we studied
the tensile strain effect on the electronic structure and carrier
transport of GeS monolayer. When the tensile strain was applied
in the zigzag direction, the indirect–direct–indirect transition can
be triggered at the strain of 3.5 and 9%, respectively. Moreover,
the HSE06 band gap of GeS monolayer was tuned from 1.96
to 2.72 eV under the uniaxial and biaxial tensile strain in the
range of 10%. Finally, upon applying external strain, the electron
mobility of GeS monolayer had an enhancement with more
than two orders of magnitude. For example, the electron mobility
had a high value of 8.11 × 105
cm2
V−1
s−1
in the armchair
direction when the tensile strain applied in the zigzag direction
reaches up to 6%. These results provide an effective way to tune
the electronic properties of GeS monolayer in electronic and
optoelectronic applications via external strain.
5. Acknowledgments: This work was financially supported
by the National Natural Science Foundation of China (grant nos.
61275108, 50672087, 50872123, and 50802083) and the Natural
Science Foundation of Zhejiang Province in China (grant no.
Y111049).
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Micro Nano Letters, 2018, Vol. 13, Iss. 5, pp. 600–605
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