Supplementary Material for
Role of Strain on Electronic and Mechanical Response of
Semiconducting Transition-Metal Dichalcogenide Monolayers:
an ab-initio study
David M. Guzman and Alejandro Strachan1
School of Materials Engineering and Birck Nanotechnology Center,
Purdue University, West Lafayette, Indiana, USA
1
Corresponding Author: strachan@purdue.edu
Structure optimization
All MX2 monolayers were built based on the DFT relaxed geometries of their bulk
systems. Table S1 shows a summary of the relaxed lattice parameters and band
gaps of the bulk materials. For comparison, the experimental values have been
included
Table S1. (Color Online) Calculated and experimental lattice parameters and band bag energy of
bulk transition-metal dichalcogenides. In all cases the band gap appears to be indirect.
PBE+vdWDF
Experimental
2H
Structures
a0 (Å)
c0 (Å)
Egap (eV)
a0 (Å)
c0 (Å)
MoS2
3.21
12.44
0.85
3.16a
12.29a
1.23a
MoSe2
3.33
13.05
0.85
3.30a
12.94a
1.09a
MoTe2
3.54
14.01
0.68
3.52a
13.97a
1.00a
WS2
3.19
13.00
1.24
3.15b
12.32b
1.35b
WSe2
3.34
13.40
1.08
3.28b
12.96b
1.20b
WTe2
3.56
14.49
0.80
--
--
--
PBE+vdWDF
1T
Structures
a0 (Å)
HfS2
3.63
5.9
HfSe2
3.80
SnS2
SnSe2
Egap (eV)
Experimental
c0 (Å) Egap (eV)
a0 (Å)
c0 (Å)
Egap (eV)
1.52
3.64c
5.67c
2.34d
6.2
0.47
3.81c
6.14c
1.00d
3.67
6.05
1.57
3.63e
5.88e
1.96f
3.78
6.64
0.89
3.74e
6.14e
1.14g
(a) Ref [1]
(b) Ref [2]
(c) Ref [3]
(d) Ref [4]
(e) Ref [5]
(f) Ref [5, 6]
(g) Ref [5, 7]
2
Formation energy
The formation energy of MX2 single-layer systems, see Figure S1, was calculated
with respect to the bulk material as
Where Emonolayer and Ebulk are the total energies per formula unit of the monolayer
and bulk systems, respectively. For the bulk systems we used a van der Waals
functional to describe the nonlocal correlation energy as function of the electron
density, as described by Dion et. al.[8] and implemented in the QUANTUM
ESPRESSO code [9]. For completeness, we calculated the formation energy of
graphene (C6) and hexagonal boron nitride (h-BN), which agree with previous
calculations [10].
0.26
Formation Energy (eV/u.f.)
0.24
0.22
0.20
0.18
0.16
0.14
0.12
0.10
MoS2 MoSe2 MoTe2 WS2 WSe2 WTe2 SnS2 SnSe2 HfS2 HfSe2
C6
h-BN
Figure S1. Calculated formation energy of the studied transition-metal chalcogenide monolayers.
For comparison we include the formation energies of graphene (C6) and hexagonal boron nitride
(h-BN)
3
Electronic structure of MX2 monolayers
Figure S2 shows the electronic band dispersion and density of states of 2H
transition-metal dichalcogenide monolayers calculated including the spin-orbit
interaction. In all cases we observe a direct band gap centered at the K point in
the two-dimensional Brillouin zone. On the other hand, the spin splitting of the
valence band increases as the chalcogen atom is changed from S to Te, where
systems with M=W exhibit a higher spin splitting as compared to monolayers
consisting of M=Mo. Monolayers of WTe2 exhibit the highest spin splitting
~400meV. Degeneracy of the conduction band close to its minima is not lifted
due to spin-orbit induced spin splitting. From the partial density of states it is
observed that the valence band maxima and conduction band minima are
originated from the transition-metal atom states.
MX2 monolayers with 1T structure do not exhibit direct band gap as shown in
Figure S3. In the case of monolayers consisting of M=Sn, we observe the
valence band maxima located at ~0.3 and the conduction band minima
centered at the M symmetry point. Similarly, cases with M=Hf have the valence
band maxima center at  and the conduction band minima at M. For both, M=Sn
and Hf, the valence and conduction bands are originated from the metal atom, as
seen from the partial density of states. Contrary to the 2H monolayers, the 1T
monolayers do not exhibit a sizable spin splitting.
The origin of the observed spin-orbit induced spin splitting in 2H MX2 monolayers
has been attributed to a loss of inversion symmetry when the dimensionality is
reduced from bulk 2H MX2 to the monolayer [11].
4
Figure S2. (Color Online) Calculated electronic band structure and partial density of states of
transition-metal dichalcogenide monolayers with trigonal prismatic coordination. The spin-orbit
interaction has been taken into account in the calculation. The Fermi energy is at 0 eV.
5
Figure S3. (Color Online) Calculated electronic band structure and partial density of states of
transition-metal dichalcogenide monolayers with octahedral coordination. The spin-orbit
interaction has been taken into account in the calculation. The Fermi energy is at 0 eV
6
Unstrained band alignment of MX2 monolayers
For completeness we also carried out the band alignment calculation without
spin-orbit interaction. We observe that the spin-orbit interaction affects the
energetics of the VBM and CBM; specifically, the absolute position of the VBM is
consistently higher when the SO interaction is turned on in the calculation, while
the CBM remains basically unchanged. Moreover, the spin-orbit interaction is
stronger in systems lacking inversion symmetry (P-6m2) than in centrosymmetric
(P-3m1) structures[11]. The analysis presented in this work is based solely on
the calculations with spin-orbit interaction.
Figure S4. (Color Online) Band Alignment of MX2 single-layer systems categorized by crystal
structure. The black solid line represents the calculated band alignment with spin-orbit interaction,
and the green dashed line represents the band alignment without spin-orbit interaction. The
vacuum level was taken as 0eV. Those compounds with asterisk (*) have been produced
experimentally as monolayers.[12, 13]
7
Role of strain on the band alignment of MX2 monolayers
Figure S5. Band structure with SO interaction for transition-metal dichalcogenide single-layer
systems with an octahedral coordination.
8
Figure S6. Band structure with SO interaction for transition-metal dichalcogenide single-layer
systems with a trigonal prismatic coordination. Systems shown consists of Mo(S,Se,Te) 2.
9
Figure S7. Band structure with SO interaction for transition-metal dichalcogenide single-layer
systems with a trigonal prismatic coordination. Systems shown consists of W(S,Se,Te)2.
10
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
Boker, T., et al., Band structure of MoS2, MoSe2, and alpha-MoTe2: Angleresolved photoelectron spectroscopy and ab initio calculations. Physical
Review B, 2001. 64(23): p. 11.
Schutte, W.J., J.L. Deboer, and F. Jellinek, CRYSTAL-STRUCTURES OF
TUNGSTEN DISULFIDE AND DISELENIDE. Journal of Solid State Chemistry,
1987. 70(2): p. 207-209.
Julien, C., et al., OPTICAL AND ELECTRICAL CHARACTERIZATIONS OF SNSE,
SNS2 AND SNSE2 SINGLE-CRYSTALS. Materials Science and Engineering BSolid State Materials for Advanced Technology, 1992. 15(1): p. 70-72.
Margaritondo, G., et al., CONDUCTION AND VALENCE BAND DENSITY OF
STATES OF SNS2 - THEORY AND EXPERIMENT. Solid State Communications,
1977. 22(12): p. 753-757.
Greenawa.Dl and R. Nitsche, PREPARATION AND OPTICAL PROPERTIES OF
GROUP 4-62 CHALCOGENIDES HAVING CDI2 STRUCTURE. Journal of Physics
and Chemistry of Solids, 1965. 26(9): p. 1445-&.
Shepherd, F.R. and P.M. Williams, PHOTOEMISSION STUDIES OF BAND
STRUCTURES OF TRANSITION-METAL DICHALCOGENIDES .1. GROUP-IVA AND
GROUP-IVB. Journal of Physics C-Solid State Physics, 1974. 7(23): p. 44164426.
Zheng, X.G., H. Kuriyaki, and K. Hirakawa, ELECTRICAL ANISOTROPY OF
LAYERED COMPOUND ZRSE2 AND HFSE2. Journal of the Physical Society of
Japan, 1989. 58(2): p. 622-626.
Dion, M., et al., Van der waals density functional for general geometries (vol 92,
art no 246401, 2004). Physical Review Letters, 2005. 95(10): p. 1.
Giannozzi, P., et al., QUANTUM ESPRESSO: a modular and open-source
software project for quantum simulations of materials. Journal of PhysicsCondensed Matter, 2009. 21(39): p. 19.
Zhuang, H.L.L. and R.G. Hennig, Electronic structures of single-layer boron
pnictides. Applied Physics Letters, 2012. 101(15): p. 4.
Zhu, Z.Y., Y.C. Cheng, and U. Schwingenschlögl, Giant spin-orbit-induced spin
splitting in two-dimensional transition-metal dichalcogenide semiconductors.
2011. 84(15): p. 153402.
Gordon, R.A., et al., Structures of exfoliated single layers of WS2, MoS2, and
MoSe2 in aqueous suspension. Physical Review B, 2002. 65(12).
Kirmayer, S., et al., Self-assembled lamellar MoS2, SnS2 and SiO2
semiconducting polymer nanocomposites. Philosophical Transactions of the
Royal Society a-Mathematical Physical and Engineering Sciences, 2007.
365(1855): p. 1489-1508.
11