We investigated the structural and electronic properties of neutral MoS2 monolayer
nanoflakes with stoichiometry MonS2n using DFT in gaussian09 [142]. In exper-
iments, usually triangular shaped islands of MoS2 have been reported but it has
been theoretically speculated that MoS2 islands can exist in various shapes, such
as trigonal, hexagonal, truncated hexagonal and rhombohedral [154, 155, 156, 157].
We used rhombic flakes shown in Fig. 4.1 to maintain the neutrality and MonS2n
stoichiometry of the flakes.
To choose the appropriate functional for modelling these small-sized nanoflakes, we made a comparison of the HOMO-LUMO gap using different functionals in gaussian09 as shown in Table 4.1. We faced an energy convergence issue when using the BP86 [108, 118] functional and did not use it for further modelling as we suspected that the convergence issues would be worse for larger flakes using this functional. For HSEH1PBE [112, 113, 114, 115, 116, 117], B3LYP [107, 108], PBE1PBE [109], B3PW91 [108, 119], PBEh1PBE [110], and M05 [102], we obtained
gaps smaller than the known experimental band gap in infinitely large sheet of MoS2
monolayer. We expect the HOMO-LUMO gap to decrease with increasing flake size
and then converge to the infinite monolayer MoS2 band gap for larger flakes. Thus
for these functionals, we expect the results to get worse with any increase in flake Table 4.1: An analysis of the HOMO-LUMO gap in gaussian09 for a 9-atom
nanoflake under different functionals.
Functionals HOMO-LUMO gap (eV)
B3LYP 0.75 BHandHLYP 3.06 HSEH1PBE 0.25 BP86 Convergence error B3PW91 1.44 PBE1PBE 1.73 PBEh1PBE 1.70 M05 0.67 M052X 3.27 42
4.2 Methodology
Table 4.2: Mean displacement, ∆R, of atoms in the central zone of an optimized
72-atom flake from the bulk experimental positions of MoS2 using several
functionals in gaussian09. All functionals except B3LYP predict mean displacements less than 5% from the bulk values.
Functionals ∆R (Å)
PBE1PBE 0.0256
M052X 0.0330
BHandHLYP 0.0400
B3LYP 0.0565
size. The M052X [103] and BHandHLYP [111] functionals predicted reasonable gaps for this small nanoflake and we can conjecture that they might asymptote near the experimental value for larger flakes. This helped us to obtain a reasonable subset of functionals to use for further modelling.
We further investigated the structural parameters using functionals listed in Table 4.2. We relaxed the nanoflakes of all sizes with BHandHLYP and M052X as they predicted reasonable HOMO-LUMO gaps along with the commonly used B3LYP and PBE1PBE functionals. We picked a relaxed 72-atom flake as this was the largest size we could model with the B3LYP functional. We compared the relative atomic positions of each atom in the central zone of the 72-atom flake with the bulk structure
[158]. The displacement ∆Ri of each atom from the bulk position is defined as
∆Ri ≡q(Xopti − Xbulk)
2+ (Y
opti − Ybulk)
2+ (Z
opti− Zbulk)
2, (4.1)
where i indexes the atoms in the central zone of the 72-atom flake. The mean value of ∆Ri, i.e., ∆R for each functional is given in Table 4.2. All functionals except B3LYP [100, 107, 108] result in less than 5% average variation from the bulk atomic positions. This indicates that the three functionals, BHandHLYP [111], PBE1PBE [109], and M052X [103] predict similar structures at similar levels of accuracy. We computed the HOMO-LUMO gap as function of flake size for all these functionals as shown in Fig. 4.2(a). We expect the HOMO-LUMO gap to decrease with increasing
flake size, approaching the experimental monolayer MoS2 gap for larger flakes. This
trend is also reported by Gan et al. [159] through an analytical equation given below
for MoS2 monolayer quantum dots of size from 2 nm to 10 nm.
E∗ = Eg+ h 2 8µr2 − 1.8q2 e 4ε εr (4.2)
4 Size-dependent properties of MoS2 monolayer nanoflakes 0 2 4 6 8 10 2 4 6 8 Size of nanoflakes (nm) HOMO-LUMO (eV) Er E∗ BHandHLYP M052X 0 40 80 120 0 1 2 3 Number of atoms HOMO-LUMO (eV) B3LYP BHandHLYP M052X PBE1PBE
Experimental monolayer gap
(a) (b)
Figure 4.2: Size-dependent analysis of the HOMO-LUMO gap in MoS2 monolayer
nanoflakes. (a) Our computed HOMO-LUMO gap using four different functionals. The HOMO-LUMO gap decreases with an increase in the size of flakes. (b) Analytical gaps calculated from the equations explained in the main text and our DFT computed gaps using BHandHLYP and M052X functionals as a function of nanoflakes sizes. The physical trends of analytical and our computed results agree with each other for nanoflakes of size up to 2 nm. The black-dashed line is the known experimental gap
in a large sheet of MoS2 monolayer [7].
Here µ is 0.16mo, the reduced mass of an exciton, mo is the free-electron mass, h is
Planck’s constant, qe is electron charge, Eg = 1.29 eV is the bulk band gap, ε = 6.8
is the relative dielectric constant [159], and r is the radius (in units of meter) of the quantum dots. An alternative analytic prediction for the HOMO-LUMO gap of a square-lattice is reported by Li et al. in [160], which we have re-solved for a circular lattice while ignoring the spin-orbit coupling term (as spin-orbit term is relatively small being of order of a tenth of an electron volt [160]). This equation is given as
Er = v u u ta2t2 ρ o r 2 + ∆ 2 !2 , (4.3)
where a = 3.193 Å is the lattice parameter of a MoS2 monolayer unit cell, t = 1.1
eV is the hopping term between the nearest neighbours, ∆ = 1.66 eV is the direct
band gap, ρo = 2.4048 is the first root of zeroth-order Bessel function, and r is the
radius of the circular lattice as defined earlier for Eq. 4.2. We have plotted the above two analytical expressions along with our DFT modelled HOMO-LUMO gap using BHandHLYP and M052X functionals for nanoflakes of various sizes as shown in 44
4.2 Methodology Fig. 4.2(b). Although our flakes are smaller than 2 nm and we are modelling in DFT, nevertheless we expect a similar trend of approximately decreasing band gap with increasing flake size. We also do not expect the energies to diverge in the limit of very small-sized nanoflakes, hence our computed values appear to be more realistic than the analytical results. Due to the different methods involved, we only compare the trends, not the absolute values of the HOMO-LUMO gaps.
Due to the known analytical results reported by Gan et al [159], we expect the HOMO-LUMO gaps to decrease with the increasing flake size and converging to the experimental values for larger flakes. As the smaller flake’s HOMO-LUMO gap values for B3LYP and PBE1PBE functionals are well below the known experimental
gap for an infinitely large MoS2 monolayer [Fig. 4.2(a)], we expect the results to
worsen as the size of flakes will grow. Hence, we do not consider these two functionals further. For smaller flakes, BHandHLYP and M052X both produce HOMO-LUMO gaps well above the monolayer experimental value [7] and we can expect the band gap with these functionals to converge close to the experimental monolayer band gap for larger flakes. Cramer and Truhlar report that M052X is not a recommended functional for transition metal chemistry [161]. Considering this, we therefore used the BHandHLYP functional in our article [162], but here we present all the results with both BHandHLYP and M052X functionals.
The hybrid DFT functional, BHandHLYP [111], includes a mixture of Hartree-Fock exchange with the DFT exchange-correlation via the relation
BHandHLYP : 0.5EHF
x + 0.5ExLSDA+ 0.5∆ExBecke88+ EcLYP; (4.4)
EHF
x is the Hartree-Fock exchange term, ExLSDA is the Slater local exchange term
[163], ∆EBecke88
x is Becke’s 1988 [100] gradient correction to the local-spin density
approximation (LSDA) for the exchange term, and ELYP
c is the Lee-Yang-Parr
correlation term [107].
M052X is a meta-GGA (generalized-gradient approximation) functional having 56% Hartree-Fock exchange term [103].
The basis set used was an effective-core potential basis set of double-zeta quality, the Los Alamos National Laboratory basis set, LANL2DZ [164] and developed by Hay and Wadt [125, 126, 127]. Hay-Wadt pseudopotential has been tested and optimized for Mo atoms and incorporate the relativistic effects. LANL2DZ basis set has been reported to be the best known for the atoms beyond third row of the periodic table
4 Size-dependent properties of MoS2 monolayer nanoflakes
[165]. Further, this basis set has been tested by Zakharov et al. [166] for Mo12S24
macromolecule in gaussian programme and they found it to perform well for this case. This basis set has also been tested by Yang et al. [167] on transition metal complexes. LANL2DZ basis sets are widely used in the study of quantum chemistry, particularly for heavy elements [164].
gaussian09 optimization criteria: calculations were converged to less than 4.5×10−3
Hartree/Bohr maximum force, 3×10−4 Hartree/Bohr RMS force, 1.8×10−3 Hartree
maximum displacement and 1.2×10−3Hartree RMS displacement. All the flakes were
converged to the default SCF (self-consistent field) limit of < 10−8 RMS change in
the density matrix except those specified in the next section. The charge multiplicity (net charge) was 0 and the spin multiplicity was 1 (singlet; spin neutral).
In the geometry optimization process, the geometry was modified until a stationary point on the potential surface was found. Analytic gradients were used and the optimization algorithm was the Berny algorithm using gediis [168]. We calculated the electronic properties of the optimized structures. The charge densities were plotted in avogadro [169, 170] from a compatible gaussian09 checkpoint file. To investigate the optical properties, we used TD-DFT (time-dependent density- functional theory) [88, 92] and configuration-interaction (CI) singles, CIS [171, 172, 173] in gaussian09. In the CIS approach, orbitals of Hartree-Fock solutions are used to generate all singly-excited determinants of the configuration-interaction expansion. TD-DFT is the most popular way to treat the excited states problem in the framework of DFT. We compared the results of both theories and solved the excited state problem in an ultraviolet to visible (UV-Vis) spectrum. To obtain deep insight in the absorption spectrum, we used the programme multiwfn [174]. We used the Gaussian broadening function with full-width half maximum (FWHM) parameter set at default value of 0.66667 eV.