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Strain dependence of band gaps and exciton energies in pure and mixed transition-metal dichalcogenides

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Abstract

The ability to fabricate 2D device architectures with desired properties, based on stacking of weakly (van der Waals) interacting atomically thin layers, is quickly becoming reality. In order to design ever more complex devices of this type, it is crucial to know the precise strain and composition dependence of the layers' electronic and optical properties. Here, we present a theoretical study of these dependences for monolayers with compositions varying from pure MX2 to the mixed MXY, where M=Mo, W and X,Y=S, Se. We employ both density-functional-theory and GW calculations, as well as values of the exciton binding energies based on a self-consistent treatment of dielectric properties, to obtain the band gaps that correspond to optical or transport measurements; we find reasonable agreement with reported experimental values for the unstrained monolayers. Our predictions for the strain-dependent electronic properties should be a useful guide in the effort to design heterostructures composed of these layers on various substrates.
PHYSICAL REVIEW B 94, 155310 (2016)
Strain dependence of band gaps and exciton energies in pure and mixed
transition-metal dichalcogenides
Rodrick Kuate Defo,1Shiang Fang,1Sharmila N. Shirodkar,2Georgios A. Tritsaris,2
Athanasios Dimoulas,3and Efthimios Kaxiras1,2
1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
2John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
3Institute of Nanoscience and Nanotechnology, National Center for Scientiﬁc Research Demokritos, 15310, Aghia Paraskevi, Athens, Greece
(Received 6 June 2016; revised manuscript received 30 August 2016; published 27 October 2016)
The ability to fabricate 2D device architectures with desired properties, based on stacking of weakly (van
der Waals) interacting atomically thin layers, is quickly becoming reality. In order to design ever more complex
devices of this type, it is crucial to know the precise strain and composition dependence of the layers’ electronic and
optical properties. Here, we present a theoretical study of these dependences for monolayers with compositions
varying from pure MX2to the mixed MXY,whereM=Mo, W and X,Y =S, Se. We employ both density-
functional-theory and GW calculations, as well as values of the exciton binding energies based on a self-consistent
treatment of dielectric properties, to obtain the band gaps that correspond to optical or transport measurements;
we ﬁnd reasonable agreement with reported experimental values for the unstrained monolayers. Our predictions
for the strain-dependent electronic properties should be a useful guide in the effort to design heterostructures
composed of these layers on various substrates.
DOI: 10.1103/PhysRevB.94.155310
I. INTRODUCTION
Interest in 2D materials, originally sparked by the discovery
of graphene, has been invigorated with the advent of single
layers that exhibit semiconducting properties. Transition metal
dichalcogenides (TMDCs, composition: MX2, where M=
transition metals like Nb, Ta, Mo, W, Ti, and X=chalcogens
like S, Se, Te) are a particularly attractive class of semicon-
ducting layered materials because their electronic and optical
properties can be manipulated by chemical substitutions as
well as by the number of layers and other structural fea-
tures (stacking sequences and polytypes). Moreover, TMDCs
exhibit unconventional phenomena such as topological su-
perconductivity [1] and charge density waves [2] and have
found widespread applications in varied areas like lubrication
[3], catalysis [4], photovoltaics [5], and optoelectronics [6].
The range of possible applications grows enormously by
considering heterostructures composed of various stacking
orders and relative orientations of the single layers. To mention
but a few examples, TMDCs can complement graphene in
optoelectronic or energy harvesting applications requiring thin
transparent semiconductors [7]; MoS2/graphene heterostruc-
tures have already been fabricated and applied to energy
harvesting with a photogain of over 108[8] and to nonvolatile
memory cells [9]; light emitting diodes have been constructed
based on heterostructures of hexagonal boron nitride, silicon
dioxide, silicon, graphene, WSe2, and MoS2[10].
In the applications mentioned, variable composition and
strain-induced effects could make a signiﬁcant difference on
the electronic and structural properties of the monolayer.
Even allowing for the fact that the van der Waals (vdW)
coupling between layers is weaker than covalent bonding,
their interaction is non-negligible as the dependence of the
electronic features on the number of layers indicates. In
addition, controlled changes in the chemical composition,
enabled by recent molecular beam epitaxy (MBE) growth
of TMDCs [11,12], suggest that strain can be an inherent
feature of each layer in a heterostructure. Investigations of
strain effects have revealed a transition from a direct to an
indirect band gap in the monolayer TMDCs [13,14] and
metallic behavior for very large strains of 10% [13,15]. More
pertinently, valley drift has been predicted under application
of strain in MoS2[16], as well as a shift of the electron
and hole band edges due to uniform strain [17], relevant
to applications in photovoltaic devices or the creation of
long-lived indirect excitons. The effect of twist angle on the
band structure of bilayer MoS2has also been investigated and
shown to induce widening of the band gap with twist angle
due to modulation of the interlayer coupling [18]. The effect
on the electronic properties of monolayer MoS2supported or
suspended by a silicon substrate has also been examined [19]
as well as the thickness dependence of electronic properties of
WSe2[20]. Composition also plays an important role in the
electronic properties of monolayers. For instance, temperature
and the value of xin the formula Mo1xWxTe2determine the
relative stability of the H or Tphases [21], the former being
semiconducting and the latter semimetallic [22,23]. Further,
the choice of chalcogen inﬂuences the piezoelectric properties
of the H phase TMDC monolayers [24].
We present here an investigation of strain effects on the
electronic and optical properties of the common TMDC mate-
rials MoS2, MoSe2and WS2,WSe
2and their mixed variants,
MoSSe, WSSe. The latter two structures do not as yet exist
but are interesting limiting cases of compositional variation.
Moreover, such structures could even be created, in principle,
by recently developed MBE techniques [11,12], especially if
a polar substrate were to be used. Such polar structures are
interesting because they open the possibility of creating TMDC
layers with inherent strain or polar character. Our results extend
earlier work [1417,25,26] in important ways: They provide
a comprehensive discussion of the inﬂuence of strain on the
dielectric function and hence the optical behavior of common
TMDCs and their mixed variants, and they give a detailed
account of how different components (dielectric constant,
2469-9950/2016/94(15)/155310(11) 155310-1 ©2016 American Physical Society
RODRICK KUATE DEFO et al. PHYSICAL REVIEW B 94, 155310 (2016)
electron and hole effective masses, exciton binding energies)
contribute to the overall changes in electronic structure induced
by strain. In this sense, the present results offer a broad basis for
designing heterostructures based on pure and mixed-character
TMDCs with desirable optoelectronic properties.
II. COMPUTATIONAL METHODS
We performed ﬁrst-principles density functional theory
(DFT) calculations for structural optimization using the GPAW
package [2729], which is a grid-based approach employing
the projected augmented-wave method [30]. For the exchange-
correlation energy of electrons we use the generalized gradient
approximation (GGA), as parametrized by Perdew, Burke,
and Erzenhof (PBE) [31]. A vacuum of 15 ˚
A separating the
adjacent periodic images along the direction perpendicular to
the plane was employed to simulate an isolated 2D planar
sheet. The atomic positions were relaxed until the magnitude
of Hellmann-Feynman forces was smaller than 0.01 eV/˚
Aon
each atom. The wave functions were expanded in a plane wave
basis with a cutoff energy of 400 eV, and a zone-centered grid
of 24 ×24 ×1 points was used for integrations in kspace for
both structural and excited-state calculations.
We carried out GW calculations for MoS2as the benchmark
compound to establish highly converged values and used
results from the literature for other compounds, converged
to similar accuracy. For these calculations we employed
the QUANTUM ESPRESSO package [32] and the BERKELEYGW
[33] code, to converge the conduction and valence band
quasiparticle energies to within 5 meV; the results of these
calculations and comparison to KS eigenvalues were discussed
in detail by Shiang et al. [34].
III. STRUCTURE AND STABILITY
In Table Iwe collect the results on the structural features
of the compounds studied here. The mixed compounds MXY
have features that are very close to the average of the features
of the two related pure compounds, MX2and MY2(M=Mo,
W and X,Y =S, Se). In Fig. 1we show the atomic structure
and valence electron density of MoS2, as a representative
case of the compounds studied, at zero strain and at ±3%
tensile/compressive strain. The application of strain slightly
distorts the electronic density, resulting in +1% increase of the
bond length bM-S for +3% tensile strain and 1% decrease of
bM-S for 3% compressive strain. The two chalcogen atoms
on either side of the central plane move farther apart from
each other for compressive (negative) strain and closer to each
other for tensile (positive) strain, thus partially mitigating the
effect of lattice strain on the metal-chalcogen bond distances.
The thickness dwas chosen such that outside of this distance
the charge density at a given point would be less than 10%
of its maximal value. From the different lattice constants,
we can infer that the free-standing mixed compounds MSSe
would form a thin spherical shell, with the Se on the outer
surface and the S on the inner surface, and with a radius of
curvature R=da/(aa)15 nm, for both the Mo and W
based compounds, where aand aare the lattice constants of
the corresponding pure compounds MSe2and MS2.
The cohesive energies in Table Iwere calculated using
Ecoh =EEb(M)2Eb(X), where Eis the total energy
of the relevant MX2compound per unit formula, Eb(M)isthe
total energy per atom for bulk M(taken as a BCC crystal) and
Eb(X) is the total energy per atom for bulk Se (modeled as
helices with three atoms per unit cell) or bulk S (modeled as
stacked S8rings). In the Se bulk structure, the bond length for
nearest-neighbor atoms is 2.41 ˚
A, the Se-Se-Se bond angle in
a chain is 103.9, and the minimum distance between chains is
3.53 ˚
A. For the S bulk structure, we calculate the total energy
of a gas phase S8molecule and then subtract the sublimation
enthalpy [35] to obtain the total energy of the solid phase;
in the molecule, the bond length is 2.06 ˚
A, the average bond
angle is 108.1, and the average dihedral angle is 98.6.
We obtain the in-plane stiffness of the TMDC monolayers
from the expression [36]:
I=1
A0
2Es
∂ε2,(1)
where A0is the equilibrium area of a unit cell of the monolayer
and Esis the difference between the total energy of the strained
structure and the total energy of the system at equilibrium,
expanded to second order in the components of the strain ε.
The in-plane stiffness Iis larger for shorter bonds (IMS2>
IMSSe >I
MSe2for M=Mo, W), consistent with previous
results [25,37]; our value for the in-plane stiffness of MoS2,
135.8 N/m, is well within the range of the experimental results,
180 ±60 N/m[38]. W-based compounds are stiffer compared
to Mo-based ones, since bonding orbitals are more extended
in W than in Mo leading to greater overlap between metal and
chalcogen orbitals and correspondingly stronger bonds.
TABLE I. Structural features of the pure MX2and mixed MXY TMDC compounds considered: lattice constant (a), distance along the
plane-normal direction between metal and chalcogen layers (dM-S,dM-Se ), bond lengths (bM-S,bM-Se ), thickness (d)(allin ˚
A), cohesive energies
(Ecoh) of equilibrium conﬁgurations (in eV), the in-plane stiffness I(in N/m), the dipole moment perpendicular to the plane of the 2D material,
p, (in Debye, D=0.2082 e˚
A), and the change in energy due to the presence of an Al-terminated polar AlN substrate, Ebpol,(ineV).For
the hybrid compounds, the changes in energy for the dipole oriented parallel (antiparallel) to the dipole of the substrate are indicated.
M-S dM-Se bM-S bM-Se dE
coh Ip
Ebpol
MoS23.18 1.56 2.41 6.15 2.45 135.8 0 0.79
MoSSe 3.24 1.53 1.71 2.42 2.53 6.30 2.24 125.3 0.25 0.89 (0.56)
MoSe23.32 — 1.67 — 2.54 6.45 2.10 115.3 0 0.64
WS23.18 1.57 2.42 6.16 2.26 151.4 0 0.61
WSSe 3.25 1.53 1.71 2.42 2.54 6.32 1.95 138.1 0.24 0.72 (0.40)
WSe23.32 — 1.68 — 2.55 6.48 1.71 127.4 0 0.49
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STRAIN DEPENDENCE OF BAND GAPS AND EXCITON . . . PHYSICAL REVIEW B 94, 155310 (2016)
FIG. 1. (a) Atomic structure and (b)–(d) valence electron densities of MoS2, a representative case of the compounds considered, for
(b) zero strain, (c) 3% compressive strain, and (d) +3% tensile strain. The white lines in (b)–(d) indicate the boundaries that deﬁne the
thickness dof the layer (see text for details).
For applications requiring materials with intrinsic dipole
moment, the two mixed compounds, MoSSe and WSSe, are
particularly interesting since they have a signiﬁcant dipole
perpendicular to the plane of about 0.24–0.25 Dper MSSe
unit, that is, 1/8 of the dipole moment of a water molecule.
We calculate this dipole moment by integrating the product
of the total charge density and the position vector over the
unit cell, which gives unambiguous results for the component
perpendicular to the slab whose charge density goes to zero
at some value well within the dimension of the unit cell in
this direction. We comment here on the possibility of creating
such polar structures: This may be feasible by growth on
polar substrates, like AlN, which exhibit strong polarity in
the direction perpendicular to the surface. The value of the
lattice constant for AlN is 3.11 ˚
A[39], implying a lattice
constant mismatch of about 4% between the substrate, which
is wurtzite in bulk form, and the MSSe layer. The more stable
termination for the AlN surface has Al on top [40,41]. We
obtain a rough estimate of the dipole moment of such a
surface from the bulk counterpart which gives values ranging
from 0.99–1.53 D[42,43]. Growth by MBE methods may be
well suited to promote the creation of these mixed compounds,
since the atoms arriving on the surface will likely have a
good chance of forming local structures with the proper
orientation, inﬂuenced by the dipole moment of the substrate.
The energies that would be gained with a polar AlN substrate
are given in Table I. These energies were obtained by taking
the difference between the energy of the attached MX2/AlN
or MXY/AlN structure and the energies of the isolated MX2
or MXY and AlN structures. For the AlN slab, the surface
closest to the TMDC is the Al-terminated one. We explored
the six high-symmetry relative lateral shifts to determine the
lowest energy conﬁguration as well as the distance from
the AlN slab to the TMDC layer for each structure. In
Fig. 2, we show the calculated probability of formation of
the mixed layers, using Boltzmann weights and the fractions
fM,fS, and fSe of reactant atoms, where fM=(u1)fS,
fSe =1ufS, with usome real number greater than 1, for
the MX2and MXY systems (M=Mo, W), which are given
by:
P(±)
MSSe(fS)=2fSfSe eE(±)
MSSe/kbT
f2
SeEMS2/kbT+2fSfSe(eE(+)
MSSe/kbT+eE()
MSSe/kbT)+f2
SeeEMSe2/kbT,(2)
where the sign refers to the orientation of the dipole moment
of MSSe relative to that of AlN (+for parallel, for
antiparallel orientation). The energies in the Boltzmann factors
are obtained by summing Ecoh and Ebpol from Table I.The
lowest of these energies is that of the MS2compounds so, for a
relatively high fraction of S atoms, formation of MS2is favored
at ambient temperature. For higher temperature, the energy
differences have less effect so a higher fraction of S atoms
maximizes the probability of forming the mixed compounds.
Naturally, a relatively large fraction of Matoms (ularge) leads
to a lower fraction of Se atoms maximizing the probability of
forming the mixed compounds, which may be important for
cost considerations, though if fMfSmore atoms would
be wasted. Figure 3shows the charge density difference, for
MoSSe on AlN, calculated as ρ =ρMoSSe+AlN ρMoSSe
ρAlN. Due to the relative electronegativities of the atoms at
the interface, charge accumulates in the region between the
IV. ELECTRONIC PROPERTIES
We turn next our attention to the electronic properties
of these systems. Since DFT underestimates the band gaps,
we also use GW results from the literature as well as a
self-consistent method for calculating excitonic effects to
determine the values of optical and transport gaps that can be
compared to experiment. To obtain the macroscopic dielectric
function we use the reciprocal of the G=G=0 component
of 1
GG(q), the inverse of the microscopic dielectric matrix
in reciprocal space, calculated within the random phase
approximation [44]. This formulation ensures that local ﬁeld
effects are included.
To facilitate further discussion, we show in Fig. 4the
band structures of the mixed MoSSe and WSSe compounds,
which have not been previously considered. These resemble
closely the band structures of the pure compounds, MS2and
MSe2(M=Mo, W), with relatively minor differences. By
projecting out the contributions of the atomic orbitals of
155310-3
RODRICK KUATE DEFO et al. PHYSICAL REVIEW B 94, 155310 (2016)
10 610 510 410 310 210 1100
fS
0
20
40
60
80
100
P(+)
MSSe(fS)(%)
1
0
6
1
0
5
1
0
4
1
0
3
1
0
1
0
1
1
0
0
f
S
f
f
0
2
0
4
0
6
0
8
0
10
0
P
(
+
)
MSSe
P
P
(
f
(
(
S
f
f
)
(
%
)
%%
293 K
950 K
W
Mo
FIG. 2. Probability [P(+)
MSSe(fS)], Eq. (2), of producing the MSSe
mixed compounds (M=Mo, green curves or M=W, blue curves)
with dipole moment oriented parallel to the dipole moment of an
AlN substrate as a function of the fraction of S atoms in the source,
for two different values of temperature, T=293 K and T=950 K.
The probability for antiparallel polarization [P()
MSSe(fS)] is negligibly
small and not shown here; this is expected on physical grounds
(parallel polarization is strongly favored since it minimizes the
electrostatic energy at the interface). For the plot, we have taken
u=50.
different atoms to the wave functions, we conﬁrm that the
states of the valence and conduction band extrema near high
symmetry points mainly originate from the 3p(4p) orbitals of
the S (Se) atoms and the 4d(5d) orbitals of Mo (W) atoms, as
in the pure compounds [15,34]. Variation in strain changes the
hybridization of these orbitals, and hence shifts the energies
of the relevant states, as discussed next.
Bulk TMDCs are indirect band-gap semiconductors. How-
ever, monolayers of TMDCs exhibit direct band gaps [22],
though these are sensitive to in-plane strain (see Fig. 5). The
range of strain considered here was chosen to illustrate trends;
typically accessible strains are probably conﬁned to smaller
values, though for MoS2breaking only occurs at an effective
strain of 6 to 11% [38]. Under a compressive or tensile isotropic
strain, each material in monolayer form makes a transition to
an indirect band-gap semiconductor and the gap decreases
both with compressive and tensile strain. Spin-orbit coupling
(SOC) effects play a signiﬁcant role in determining the value
of the band gap, as shown in Fig. 5, where results with and
without SOC corrections are presented for comparison. Our
calculations indicate that WSe2is the least sensitive to strain
(exhibiting robustness of the direct band gap), whereas the
lighter compounds MoS2and MoSSe are the most sensitive.
Between the Kand Qpoints there are two contributions
from SOC that conspire to decrease the gap, one from the
conduction band at the Qpoint and one from the valence band
at the Kpoint. For the direct gap at the Kpoint there is only
one contribution from the valence band, and for the transition
between the and the Kpoint there is negligible contribution,
due in part to Kramers degeneracy. Thus, compounds with
large SOC interaction exhibit a downward shift of the indirect
KQgap value (red line in Fig. 5) when these contributions
are included. A similar shift is found for the direct KK
gap (black line in Fig. 5), to a lesser extent, while this effect
is absent in the results for the indirect K gap (green line
in Fig. 5). The result is that for the heavier compounds the
direct band gap region will be extended to larger strains. This
FIG. 3. The charge density difference between the combined MoSSe-AlN system and the sum of the isolated MoSSe and AlN substrate
and the plane-averaged electron density difference (ρz) along the direction perpendicular to the interface of MoSSe-AlN with the dipole
moment of the MoSSe oriented parallel and antiparallel to the dipole moment of the substrate; red indicates charge accumulation and blue
charge depletion.
155310-4
STRAIN DEPENDENCE OF BAND GAPS AND EXCITON . . . PHYSICAL REVIEW B 94, 155310 (2016)
-12
-10
-8
-6
-4
-2
0
Ene rg y (e V)
(a) MoSSe
Γ M K Q
-20
-18
Γ
-12
-10
-8
-6
-4
-2
0
Ene rg y (e V)
(b ) WSSe
Γ M K QΓ
-20
-18
FIG. 4. Band structure plots for (a) MoSSe and (b) WSSe at zero strain; DFT values were rescaled according to our GW calculations and
those of Ramasubramaniam [45], with bands color-coded to denote their orbital character: red =s, blue =p, green =d; these results include
spin-orbit coupling (SOC). The double-headed black, red, and green arrows indicate pairs of states that deﬁne the direct or indirect band gaps,
depending on the strain; at zero strain, the band is direct (black arrow) at K.
is due to the fact that the spin-orbit interaction will shrink the
direct band gap, though not the Kgap, and therefore
it will shift the crossing point of the direct band gap and
the indirect Kgap curves towards larger positive strains.
The compounds with the largest effects of this type are MoSe2,
WSSe, and WSe2, as indicated explicitly by the downward
arrows in Fig. 5.
We investigate next the effect of strain on the component of
the dielectric function parallel to the plane of the layer, (ω).
This effect of strain on the dielectric function is important
because dielectric screening of the MoS2monolayer enhances
mobility [7] and affects excitonic binding energies, thus inﬂu-
ences the photoluminescence [46]. Knowledge of the dielectric
function is therefore of particular importance in heterostruc-
tures consisting of multiple layers of TMDCs. Generally, the
dielectric screening is optimal when Im[(ω)]/Re[(ω)] is
near zero, which implies that losses are minimal. In order to
obtain values of the dielectric function that are independent of
the size of the periodic cell which contains both the monolayer
and the vacuum region, we need to transform the real calculated
values to Re[(ω)], where
Re[(ω)] =(Re[DFT
(ω)] 1)c/d +1(3)
with cbeing the total length of the periodic unit cell in
the zdirection and dthe thickness of a single layer as
deﬁned in Fig. 1and tabulated in Table I. This scaling gives
accurate values of the dielectric constant and is obtained by
using the rules for addition of capacitance in series and in
parallel in agreement with results found in literature [47]. A
similar expression applies to the dielectric function component
perpendicular to the layer, speciﬁcally to Re[DFT
(ω)]1, and
the imaginary part of the dielectric function is rescaled by the
factor c/d.
We ﬁnd that the dielectric constant or relative permittivity
tends to increase with strain and can thus be used as a
signature to measure the strain in these compounds. We discuss
as representative examples the dielectric functions of MoS2
and WSe2, which show similar behavior, as do all the other
compounds we considered. The features labeled X1and X2
in the plots of Re[(ω)] in Fig. 6originate from the direct
transition at the Kpoint; from Fig. 5we see that these features
track closely with the direct band gap as a function of strain.
The larger-amplitude and more diffuse feature at ωbetween 2
and 3 eV traces its origin in the regions near the Qand points.
With tensile strain (increase in the lattice constant), the peaks
in the absorption spectrum or imaginary part of (ω), which
correspond to interband excitations, shift to lower energy (see
Fig. 6).
We next turn our attention to obtaining values of the
band gaps that can be directly compared to experimental
measurements. It is well established that band gap values
obtained from DFT are underestimates of those measured in
experiment. There are two main reasons for this discrepancy:
(i) differences in single-particle energies of the Kohn-Sham
equations, which are used to obtain the DFT band gaps, are
physically justiﬁable for the ground state but not for excited
states that involve moving an electron from a valence to a
conduction band across the band gap, and (ii) there is a deriva-
tive discontinuity in the exact exchange-correlation energy
at integer particle numbers [54]. To address the ﬁrst issue,
the correct excitation energies can be calculated by solving
for the quasiparticle energies from the self-energy operator
[55,56]; though this is a useful and highly accurate approach
as implemented by the GW approximation, the computations
are quite expensive. To address the second issue, various
formulas for correcting the derivative discontinuity have been
derived, that give reasonable results for 3D solids. We have
attempted to apply one of these corrections [57] but found
that they do not give satisfactory results for the TMDC single
layers considered here. We show in Fig. 7the values obtained
from GW calculations and compare them to values obtained
from transport measurements, which are quite large (in the
range of 2.4–2.8 eV); there is quite reasonable agreement
between the GW results for all the compounds considered
here. By contrast, the results from DFT calculations, that
is, the difference between the valence band maximum and
conduction band minimum of Kohn-Sham eigenvalues (in the
range 1.3–1.6 eV) is a very signiﬁcant underestimate of the
155310-5
RODRICK KUATE DEFO et al. PHYSICAL REVIEW B 94, 155310 (2016)
(c)
(d)
(a)
(b)
KS
Fitted
FIG. 5. Phase diagrams showing the transition from a direct to an indirect band-gap semiconductor for: (a) and (b) Mo-based structures and
(c) and (d) W-based structures; for comparison we show results both without [(a) and (c)] and with [(b) and (d)] SOC effects in each case. The
transitions indicate lowest band gaps, indirect (K Q, red or K, green) and direct (K K, black); color shaded regions identify the
corresponding indirect gap ranges depending on strain. Downward (black and red) and side (gray) arrows indicate the most signiﬁcant changes
introduced by the inclusion of SOC effects. The upper curves (triangles) in [(b) and (d)] are adjusted values to match GW gaps at zero strain
and assuming the same strain dependence as in the KS gaps (see text for details).
experimental band gaps, by approximately 50%, as is typical.
In order to reconcile the Kohn-Sham eigenvalues with the
GW results, we adjust the KS values by the amount they
differ from the GW result at zero strain and assume the same
strain dependence; the results are shown in Fig. 5, and can
be taken as an approximation to the true transport gap as a
function of strain. The optical gap can then be obtained from
the exciton binding energies presented in Sec. V.TheGW
corrections can be state dependent and might therefore affect
the direct and indirect band gaps differently, which will affect
the direct-gap windows shown in Fig. 5; however, the overall
trends, as dictated by the inﬂuence of SOC, should still hold.
V. EXCITONIC EFFECTS
The results discussed so far, both experiment and theory,
do not include excitonic effects, as is appropriate for transport
measurements. To account for excitonic effects, we investigate
the binding of excitons at Kvalleys, corresponding to the
zero strain case and a direct band gap for all the materials
investigated, within the effective mass approximation, using
a classical interaction potential between electron and hole
charges [58,59]. We adopt a model for the TMDC layer
consisting of a quasi-two-dimensional anisotropic dielectric
slab of thickness dwith the in-plane (out-of-plane) dielectric
constant () immersed in vacuum, or including a substrate.
We use two different models to calculate the exciton binding
energies, one in which the electron and the hole are treated
as lines that span the thickness of the layer and one in which
they are treated as point particles, as shown in the schematic
diagrams in Fig. 7. The ﬁrst model [60] takes into consideration
the fact that the electron and the hole are described by wave
functions that are likely to be nonzero through the entire
thickness of the layer. The results of this model agree with
155310-6
STRAIN DEPENDENCE OF BAND GAPS AND EXCITON . . . PHYSICAL REVIEW B 94, 155310 (2016)
FIG. 6. Real and imaginary parts of the dielectric function, Re[(ω)], Im[(ω)], for MoS2and WSe2for in-plane strain ranging from
5% to +5% in increments of 1%. The color coding reﬂects the nature of the band gap that corresponds to each strain value, with the same
conventions as in Fig. 5. The minimum direct band gaps, which signal the onset of absorption, are identiﬁed by the gray surface in the plots for
the imaginary part of the dielectric function. The dielectric function has been transformed to (ω) from Eq. (3), see text.
the strict 2D limit model often given in the literature [61].
The second model, the limit of the two charges being point
particles, is also a sensible approach since the electron and hole
wave functions are mainly composed of the relatively localized
metal dorbitals as we have shown recently [34]. The Poisson
equation for the electrostatic problem can be analytically
solved with the help of a partial Fourier transformation for
the in-plane xy coordinate [59,62]. The potential between two
1.0
1.5
2.0
2.5
3.0
En er gy (e V)
Exp(O)
Exp(T)
GW
KS
0.72
0.72
0.67
0.66
0.65
0.60
1.0
1.5
2.0
2.5
3.0
En er gy (e V)
MoS2
MoSSe
2
WS2
WSe2
Exp(O)
Exp(T)
GW
KS
0.56
0.55
0.52
0.55
0.53
0.50
MoSe
WSSe
MoS2
MoSSe
MoSe2
WS2
WSSe
WSe2
FIG. 7. Band gaps from theory without excitonic effects, KS (green circles) and GW [45] (red circles) compared to experimental transport
measurements [46,4850] (red squares), and from theory with excitonic effects (shifted black circles) obtained with the line (left panel) or
point (right panel) charge models, compared to experimental optical measurements [5153] (black squares). The arrows and corresponding
numbers indicate the calculated exciton binding energies from each model. Open squares and circles indicate that the values were calculated
as the average of the corresponding MS2and MSe2gaps.
155310-7
RODRICK KUATE DEFO et al. PHYSICAL REVIEW B 94, 155310 (2016)
charges ±ein a layer of thickness dis given by:
U(r)=e2
4π0λ
0
J0(qr)K(q)dq, (4)
where the kernel K(q) takes different forms: for the two line
charges
Kline(q)=11
(κqd/2)[λ+coth(κqd/2)] 1
(κqd/2) (5)
whereas for two point charges
Kpoint(q)=λ+tanh(κqd/2)
1+λtanh(κqd/2) ,(6)
where q=|q|for the two-dimensional in-plane momentum
and J0is the Bessel function of the ﬁrst kind and we have
deﬁned λ=()1/2and κ=(/)1/2. With this potential,
we can then solve numerically the effective mass Schr¨
odinger
equation in the relative coordinate of the electron and the hole
charged particle, using a logarithmic grid. All values that enter
in these equations, including the dielectric constants ,,
and the effective masses for the electron and hole, are obtained
from the ﬁrst-principles calculations described earlier, so the
results do not involve any adjustable parameters. The results
from both models are shown in Fig. 7. The inclusion of
excitonic effects with the line-charge model produces values
for the optical gaps that are on average 0.54 eV lower than
the transport gaps, but this is still far from the corresponding
experimental values. In contrast to this, the results from the
point-charge model are in better agreement with the optical
gaps from experiment, and involve an average binding energy
of the excitons of 0.64 eV for the W compounds and
0.70 eV for the Mo compounds. From this comparison, it
appears that the point-charge model is closer to the physical
picture of how excitons behave in these layered compounds.
We have also explored the effect of a substrate, using the
kernel:
Kpts(q)=2(λcosh(κqd/2) +sinh(κqd/2))(λcosh(κqd/2) +ssinh(κqd/2))
(1 +s)λcosh(κqd)+(s+λ2)sinh(κqd),(7)
where sis the relative permittivity of the substrate. We
investigate the dependence of the binding energy of the exciton
on the relative permittivity of the substrate in Fig. 8for
the monolayer MoS2and WS2systems for which accurate
measurements of the exciton binding energy exist. The results
of Fig. 8indicate the possible effective values of the relative
permittivity of the substrate that could explain the observed
binding energies. These effective values could be the result of
combining the relative permittivity of the pure substrates (Si,
SiO2, h-BN) in a way that reﬂects the actual composition
and structure of the substrate, about which there is not
1 2 3 4 5 6 7 8 9 10 11 12
Εs
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Eb(eV)
WS2
MoS2
h-BN SiO2Si
FIG. 8. Binding energy of the exciton as a function of relative
permittivity of the substrate s. The horizontal dashed lines indicate
experimental measurements of the binding energy [53,64]. The
measurements on monolayer WS2were done on a SiO2/Si substrate,
while those on monolayer MoS2were done on a ﬂake of h-BN on
SiO2/Si. The relative permittivity of SiO2is 3.9[65], of Si is 11.7,
andofh-BNisintherange2to4[66]; these relative permittivity
values are indicated by the shaded regions.
enough quantitative information to allow us a more precise
estimate. Additional effects due to charge accumulation at
the interface in the presence of a polar substrate may also
come into play. The transport gap will also be reduced as
a result of the polarization-induced screening effect of the
substrate. Indeed, a reduction in the GW band gap of carbon
nanotubes of 0.35 eV is observed when they are deposited
on h-BN [63]. This suggests the optical gap would remain
relatively unchanged, but detailed calculations are necessary
to determine the magnitude of this effect.
The key parameters that enter in the determination of the
exciton binding energies are the dielectric constant (,) and
the reduced mass of the electron-hole pair μ, both of which can
be changed by applying strain. To investigate this, we show
in Fig. 9the values of these quantities for various materials
as functions of strain and the resulting binding energies of
the excitons in the lowest-energy state corresponding to zero
angular momentum, as obtained from the point-charge model.
The out-of-plane dielectric constant is affected less and
varies almost linearly with strain, with values of 4.0 to 3.2
for MoS2and WS2, 4.4 to 3.4 for MoSSe and WSSe, and
4.9 to 3.7 for MoSe2and WSe2for strain in the 5% to
+5% range. The in-plane dielectric constant shows stronger
dependence on strain and on the composition of the compound,
asshowninFig.9(a). The variation of the reduced exciton
mass μwith strain is mild and is most pronounced for the
mixed compounds MXY. The resulting variation in exciton
binding energies is monotonic and almost linear with strain
for the pure compounds and somewhat stronger for the mixed
compounds, with nonlinear behavior (stronger dependence)
for compressive strain.
VI. CONCLUSION
In this paper we explored the effect of strain and com-
position on the structural, electronic, and optical properties
155310-8
STRAIN DEPENDENCE OF BAND GAPS AND EXCITON . . . PHYSICAL REVIEW B 94, 155310 (2016)
-6 -4 -2 0246
0.1
0.3
Eb(eV)
μ/me
0.5
0.7
0.9
-6 -4 -2 0 2 4 6
11
12
13
14
15
16
MoS2
MoSSe
MoSe2
WS2
WSSe
WSe2
(b)
(a)
Ε||
FIG. 9. (a) Dielectric constant values in the plane () of the monolayers MoS2,MoSSe,MoSe
2,WS
2, WSSe, and WSe2. (b) Effective
masses (μ) in units of the bare electron mass and exciton binding energies (Eb) obtained by approximating the electron and hole as point
charges.
of MXY TMDC materials, with M=Mo, W and X,Y =S,
Se. We ﬁnd that, at the level of GW calculations, the band
gaps obtained from theory are in reasonable agreement with
band gaps from transport measurements and are in the range
2.4–2.8 eV. The band gaps are sensitive to applied strain and
can change from direct to indirect, for both compressive and
tensile strain; the indirect gaps induced by strain are different
for different signs of the strain: They occur between the K
and Qpoints of the BZ for negative (compressive) strain
and between the and Kpoints for positive (tensile) strain.
Though the choice ultimately depends on the application,
WSe2is a good candidate material for optoelectronics since
the direct gap is preserved for a wide range of tensile
in-plane strain. A tensile in-plane strain is induced in most
epitaxial MX2materials during cooling from the growth
temperature down to room temperature because these materials
typically have a larger thermal expansion coefﬁcient than
conventional substrates, therefore, during cooling, the MX2in
order to comply with the substrate lattice becomes stretched.
The strain also affects optical properties, as evidenced by
a shift in peaks of the dielectric function with applied
strain.
In order to make quantitative comparisons with optically
measured gaps, we include excitonic effects within the effec-
tive mass approximation using a screened interaction potential
between electrons and holes, which are modeled in the two
physically plausible limits as either line or point charges
within each layer. For our point charge model, the exciton
corrections yield band gaps which are in good agreement with
optically measured values. We also analyze substrate effects
on the exciton binding energies since the presence of the
substrate introduces an effective relative permittivity, which
can be obtained by taking into consideration the substrate
composition and structure. The effective permittivity of the
substrate has a signiﬁcant effect on the exciton binding
energy and brings the calculated values well within range of
experimentally measured ones.
Overall, we ﬁnd that the use of strain and composition
as independent parameters for tuning the material properties
can be very effective: For example, in the case of the mixed
MoXY compound, compressive strain of 5% can lead to
exciton binding energies in excess of 0.9 eV. The information
from our calculations makes it feasible to identify the material
with the desired value of the band gap over a range which is
considerably extended over the inherent values in strain-free
layers. Speciﬁcally, the band-gap range for the pure and mixed
compounds we considered can be from 1.5 to 3.1 eV for the
transport gap and from 1.0 to 2.1 eV for the optical gap, by
the proper combination of strain, compound composition, and
substrate choice.
ACKNOWLEDGMENTS
We thank Dmitry Vinichenko and Wei Chen for useful
discussions. We acknowledge support by ARO MURI Grant
No. W911NF-14-1-0247 (S.N.S., G.A.T., and E.K.) and by
the STC Center for Integrated Quantum Materials, NSF Grant
No. DMR-1231319 (S.F.).
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... On the other hand, Janus structures, a novel member of the family of 2D materials, have recently gained a growing interest on account to their asymmetric nanostructure, inherent out-of-plane polarization, and piezoelectricity [71,72]. In this way, Lu et al. [73] has successfully synthesized a one-layer thick Janus MoSSe by thermally selenizing the top S layer of MoS 2 with Se atoms transforming accordingly the D 3h space group of 2H-MX 2 to C 3v for Janus MXY [74,75]. Zhang et al. [76] has also prepared a SMoSe Janus monolayer by substituting with S elements the upper layer of Se atoms in MoSe 2 . ...
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... Due to the underestimation, we switched to HSE and found the bandgap for MoS 2 and ZnO as 2.21eV and 3.30eV respectively in accordance with literature [27]. The lattice constant of MoS 2 and MoSe 2 is not changed and the lattice consatnt of ZnO is varied for constructing the HS due to sensitivity of the electronic properties of MoS 2 to strain [44]. In the MoS 2 /ZnO, the lattice mismatch is 3.65% ...
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