by Da Zhou
Abstract
In this post, ab initio density functional theory (DFT) calculations with different functionals and spin-orbit coupling (SOC) parameters were performed to study the band structure of bulk and monolayer WSe2.
Introduction
Recently, more and more research interest has been drawn into the monolayer transition metal dichalcogenides (TMDs) for their novel physical properties and potential device applications. For example, the bulk MX2 (M=Mo, W; X=S, Se) has indirect bandgaps whereas the monolayer MX2 has direct bandgaps located at the K points. Therefore it is highly desirable to use DFT to numerically calculate the band structure of the TMDs and compare the calculation results with the experimental ones. Here in this post, the band structure of bulk and monolayer WSe2 were numerically calculated by the plane-wave based CASTEP, and local density approximation (LDA) and generalized gradient approximation (GGA) exchange-correlation functionals were used.
Method details
For the bulk WSe2, the lattice constants were 3.327 angstroms in a/b axis and 15.069 in c axis (the data was directly imported from materialsproject.org and no relaxation was performed). The monolayer slab was cleaved in the 001 direction and a vacuum layer of 20 angstroms was added to eliminate interlayer interactions.
Figure 1a. A ball and stick model of one unit cell of the bulk WSe2. The blue atoms represent the W atoms and the yellow atoms represent the Se atoms.
Figure 1b. A ball and stick model of the monolayer WSe2. The blue atom represents the W atom and the yellow atoms represent the Se atoms.
Based on the work by Julia Gusakova et al [1], 50 Ry which is about 680eV was enough for the energy cutoff. The energy cutoff for all of the calculations in this post was 700eV. For both bulk and monolayer WSe2 band structure calculations, a sampling separation of 0.015 1/angstrom was used. SCF tolerance was 1.0e-5 eV/atom and electronic minimizer was all bands/EDFT. Usually, the density mixing option is more recommended for the choice of electronic minimizer. But to perform calculations with the SOC effect included, all bands/EDFT were required. For the purpose of consistency, the calculations without SOC also used all bands/EDFT as the electronic minimizer to make sure the comparison of results being meaningful. The Monkhorst-Pack k-point set was 15 by 15 by 3 for the bulk and 16 by 16 by 2 for the monolayer. And norm-conserving pseudopotentials were used. The pseudo atomic calculation performed for Se: 4s2 4p4. The pseudo atomic calculation performed for W: 5d4 6s2.
Results
A series of calculations were performed on the bulk and monolayer WSe2, with LDA CAPZ or GGA PBE functionals and also with or without SOC. Only the two bands closet to the Fermi energy were plotted in the figures. And the results were summarized at the table below.
It is quite exciting to see that all the calculations, other than the one with GGA PBE functional and without SOC, successfully captured the indirect(direct) bandgap feature for the bulk(monolayer). It is also noticeable that after including SOC, the bandgap tends to be about 0.3eV smaller than the bandgap without SOC for both bulk and monolayer, which makes the calculated band gap values for the bulk very close to the experimental reference[1]. For the monolayer, the results will not be discussed here in this post since there is the experimental controversy over its bandgap being direct or indirect[5], and also the value of its bandgap varies from different measurements[1][6].
Conclusion
Most GGA and LDA calculated band structures matched well with the traditional experimental reference in terms of the bulk WSe2 having indirect bandgap and the monolayer WSe2 having direct bandgap located at the K points. And within the numerical accuracy, the calculated band gap values for the bulk were very close to the reference value when SOC was included. The work can be more meaningful if more exchange-correlation functionals can be used, and if the experimental reference results for the monolayer WSe2 can converge. Nonetheless, the calculated band structures of the monolayer WSe2 in this post are all direct bandgaps, which certainly supports one kind of opinion.
References
[1] Julia Gusakova et al, Electronic Properties of Bulk and Monolayer TMDs: Theoretical Study Within DFT Framework (GVJ‐2e Method), Phys. Status Solidi A 2017, 214, 1700218
[2] Hohenberg, P. and Kohn, W., Inhomogeneous electron gas, Phys. Rev. 1964
[3] Kohn, W. and Sham, L. J., Self-consistent equations including exchange and correlation effects, Phys. Rev. 1965
[4] Clark, S. J., and Segall, M. D., and Pickard, C. J. and Hasnip, P. J. and Probert, M. J. and Refson, K. and Payne, M.C., First-principles methods using {CASTEP}, Z. Kristall. 2005
[5]Hsu, W., Lu, L., Wang, D. et al. Evidence of indirect gap in monolayer WSe2. Nat Commun 8, 929 (2017)
[6]Zhao, W., Huang, Y., Shen, C. et al. Electronic structure of exfoliated millimeter-sized monolayer WSe2 on silicon wafer. Nano Res. 12, 3095–3100 (2019)
