GW band structure of monolayer ${\mathrm{MoS}}_2$ using the SternheimerGW method and effect of dielectric environment

Monolayers of transition-metal dichalcogenides (TMDs) hold great promise as future nanoelectronic and optoelectronic devices. An essential feature for achieving high device performance is the use of suitable supporting substrates, which can affect the electronic and optical properties of these two-dimensional (2D) materials. Here, we perform many-body GW calculations using the SternheimerGW method to investigate the quasiparticle band structure of monolayer ${\mathrm{MoS}}_2$ subject to an effective dielectric screening model, which is meant to approximately describe substrate polarization in real device applications. We show that, within this model, the dielectric screening has a sizable effect on the quasiparticle band gap; for example, the gap renormalization is as large as 250 meV for ${\mathrm{MoS}}_2$ with model screening corresponding to ${\mathrm{SiO}}_2$. Within the ${\mathrm{G}}_0{\mathrm{W}}_0$ approximation, we also find that the inclusion of the effective screening induces a direct band gap, in contrast to the unscreened monolayer. We also find that the dielectric screening induces an enhancement of the carrier effective masses by as much as 27% for holes, shifts plasmon satellites, and redistributes quasiparticle weight. Our results highlight the importance of the dielectric environment in the design of 2D TMD-based devices.

Figure 1

(a) and (b) Schematic of a ${\mathrm{MoS}}_2$ monolayer on h-BN and ${\mathrm{SiO}}_2$ substrates. In the present work the substrate is modeled using an effective dielectric environment, as shown in (c). The Coulomb interaction between two point charges with charge $e$ sitting at the distance $d$ right at the interface is $e^2/4\pi {\varepsilon }_0{\varepsilon }_{\mathrm{eff}}$, where ${\varepsilon }_0$ is the permittivity of the vacuum and ${\varepsilon }_{\mathrm{eff}}=({\varepsilon }_1+{\varepsilon }_2)/2$. The derivation of this result can be found in Sec. 4.4 of Ref. [87], among others.

Figure 2

Difference $\mathrm{\Delta }E$ of the quasiparticle band gap (QP gap), valence band maximum (VBM), and conduction band minimum (CBM) from the corresponding converged values as a function of (a) exchange ($E_{\mathrm{x}}$) and (b) correlation ($E_{\mathrm{c}}$) self-energy cutoffs. The values are obtained at the high-symmetry $\mathrm{K}$ point. The differences between the last two values of the gap in (a) and (b) are 2 and 16 meV, respectively.

Figure 3

(a) ${\mathrm{G}}_0{\mathrm{W}}_0$ (red) and DFT (indigo) band structures of monolayer ${\mathrm{MoS}}_2$. The origin of the energy axis is set to the VBM at the $\mathrm{K}$ point. The ${\mathrm{G}}_0{\mathrm{W}}_0$ band structures are calculate within the PPA, using the dielectric screening model corresponding to a ${\mathrm{SiO}}_2$ substrate. (b) Highest valence band and lowest conduction band calculated within ${\mathrm{G}}_0{\mathrm{W}}_0$/PPA, highlighting the change in the band gap character from indirect to direct when moving from the unscreened case [freestanding (FS) blue] to screening by an ${\mathrm{SiO}}_2$ substrate (red). The CBMs at $\mathrm{K}$ have been aligned for comparison.

Figure 4

Quasiparticle band gap of a ${\mathrm{MoS}}_2$ monolayer on different substrates reported in the literature. The experimental results shown by blue triangles were obtained with scanning tunneling microscopy/spectroscopy [28, 29, 31, 33, 51, 52, 53, 93, 94, 95, 96, 97, 98], absorbance [31], and angle-resolved (inverse) photoemission spectroscopy [32]. The GW band gaps are shown by the orange disks [17, 55, 99, 100]. The horizontal lines represent our calculated quasiparticle gaps using FF integration for the unscreened monolayer [freestanding (FS)] and for h-BN and ${\mathrm{SiO}}_2$ screening. The GW calculations of (a) Ref. [100], (b) Ref. [99], (c) Ref. [17], and (d) Ref. [55] reported in the plot were also obtained by considering models to describe the effective environmental dielectric screening.

Figure 5

(a) and (b) Real part of the ${\mathrm{G}}_0{\mathrm{W}}_0$ self-energy $\mathrm{\Sigma }$ of monolayer ${\mathrm{MoS}}_2$ for the VBM and CBM states. (c) and (d) Corresponding imaginary part of the self-energy. (e) and (f) Corresponding spectral functions $A(\omega ,k$). All calculations are performed at the $\mathrm{K}$ point for the unscreened monolayer [freestanding (FS), black], the case with h-BN screening (red), and the case with ${\mathrm{SiO}}_2$ screening (blue).