Substrate screening effects on the quasiparticle band gap and defect charge transition levels in ${\mathrm{MoS}}_2$

Monolayer ${\mathrm{MoS}}_2$ has emerged as an interesting material for nanoelectronic and optoelectronic devices. The effect of substrate screening and defects on the electronic structure of ${\mathrm{MoS}}_2$ are important considerations in the design of such devices. We find a giant renormalization to the free-standing quasiparticle band gap in the presence of metallic substrates, in agreement with recent scanning tunneling spectroscopy and photoluminescence experiments. Our sulfur vacancy defect calculations using the density functional theory plus GW formalism, reveal two charge transition levels (CTLs) in the pristine band gap of ${\mathrm{MoS}}_2$. The $(0/-1)$ CTL is significantly renormalized with the choice of substrate, with respect to the pristine valence band maximum (VBM). The $(+1/0)$ level, on the other hand, is pinned 100 meV above the pristine VBM for the different substrates. This opens up a pathway to effectively engineer defect charge transition levels in two-dimensional materials through the choice of substrate.

Figure 1

(a) Single-layer ${\mathrm{MoS}}_2$ on a graphene substrate, top view and side view. The black solid line marks the unit cell of ${\mathrm{MoS}}_2$. The dotted line marks the lattice-matched supercell used to perform the DFT calculations. (b)–(d) DFT band structures of free-standing $5\times 5$ supercell of graphene, lattice-matched graphene-${\mathrm{MoS}}_2$ heterostructure, and free-standing $4\times 4$ supercell of ${\mathrm{MoS}}_2$, respectively. The colors indicate the projected weights of the heterostructure wave functions onto the free-standing layers. (e) The charge density of the ${\mathrm{MoS}}_2$-graphene heterostructure, in $e/{\AA }^3$, averaged along one of the in-plane lattice directions. (f) The charge density difference, between the ${\mathrm{MoS}}_2$-graphene heterostructure and the corresponding free-standing layers. (g) The potential of the ${\mathrm{MoS}}_2$-graphene heterostructure (red line), the free-standing graphene, and the free-standing ${\mathrm{MoS}}_2$ layer (blue line).

Figure 2

(a) Quasiparticle band gap of ${\mathrm{MoS}}_2$, free-standing, and in the presence of monolayer BN, bulk ${\mathrm{SiO}}_2$, graphene (G), bilayer graphene (BLG), trilayer graphene (TLG), and graphite substrates. Experimental measurements of the quasiparticle gap in these systems are also shown in the plot. (b) Quasiparticle band gap of ${\mathrm{MoS}}_2$ in the presence of graphene as a function of increasing interlayer spacing $d_{\mathrm{G}}$.

Figure 3

(a) DFT computed band structure of a sulfur vacancy defect in a $5\times 5$ supercell of ${\mathrm{MoS}}_2$. Three defect states are induced in the gap. The filled defect level is indicated in green, and the unfilled levels are doubly degenerate and indicated in blue. The black dashed line marks the Fermi level. Partial density of states (DOS) of the $5\times 5$ supercell of ${\mathrm{MoS}}_2$ with a sulfur vacancy is on the right. The red line shows the Mo-$d$ contribution and the green line shows the S-$p$ contribution to the total density of states (black). The density of states are in units of states/eV/supercell. (b) and (c) Isosurface of the defect levels induced in the gap of ${\mathrm{MoS}}_2$. The wave function plotted in blue is the corresponding unfilled defect level in the band structure and the one plotted in green is the filled defect level. The top view as well as the side view are shown.

Figure 4

(a) Schematic of the DFT prisitine VBM, pristine CBM, and the defect levels, plotted with respect to the vacuum level. (b) Schematic of the GW pristine VBM, pristine CBM, and the defect levels, plotted with respect to the vacuum level. The dotted line is a guide to the eye, showing that the unfilled defect levels line up between DFT and GW levels of theory.

Figure 5

Formation energy of the sulfur vacancy in different charge states as a function of the Fermi level. The Fermi level is taken to scan the energy range between the pristine VBM and CBM. (a) and (b) Computed at the DFT level, for sulfur-rich and sulfur-poor conditions, respectively. (c) and (d) Computed using the DFT+GW formalism, for sulfur-rich and sulfur-poor conditions, respectively. The charge transition levels that appear in the gap are marked with red dashed lines.

Figure 6

The ${\varepsilon }^{+1/0}$ and ${\varepsilon }^{0/-1}$ charge transition levels of the sulfur vacancy defect computed using the DFT+GW formalism. The levels are shown with respect to the valence band maximum of pristine ${\mathrm{MoS}}_2$ in the presence of BN, silica, graphene (G), bilayer graphene (BLG), and graphite (Gr) substrates.