Influence of the effective layer thickness on the ground-state and excitonic properties of transition-metal dichalcogenide systems

A self-consistent scheme for the calculations of the interacting ground state and the near band-gap optical spectra of mono- and multilayer transition-metal-dichalcogenide systems is presented. The approach combines a dielectric model for the Coulomb interaction potential in a multilayer environment, gap equations for the renormalized ground state, and the Dirac-Wannier equation to determine the excitonic properties. To account for the extension of the individual monolayers perpendicular to their basic plane, an effective thickness parameter in the Coulomb interaction potential is introduced. Numerical evaluations show that the resulting finite thickness effects lead to significant modifications in the optical spectra, reproducing the experimentally observed nonhydrogenic features of the excitonic resonance series. Applying the theory for a variety of experimentally relevant configurations, a consistent description of the near band-gap optical properties is obtained all the way from monolayer to bulk. In addition to the well-known in-plane excitons, also interlayer excitons occur in multilayer systems suggesting a reinterpretation of experimental results obtained for bulk material.

Figure 1

Schematic of the model system. The distance between the van der Waals bonded layers is denoted by $D$.

Figure 2

Renormalized band gap energy (left) and Fermi-velocity (inset) distributions for $d=1.0{\lambda }_C$ and $\alpha =1.0$ (green), 3.0 (cyan), and 5.0 (blue). The resulting renormalized single-particle dispersion is shown on the right. The black dotted lines represent the noninteracting ground state properties.

Figure 3

Left: Renormalized gap as function of coupling constant $\alpha$ for three different thickness parameters $d=0.5{\lambda }_C$, $1.0{\lambda }_C$, and $2.0{\lambda }_C$. Right: Dependence of the renormalized gap on the effective thickness parameter $d$ for three values of the coupling constant $\alpha =1.0$, 3.0, and 5.0.

Figure 4

Binding energy of the $1s$ and $2s$ exciton in dependence of the effective thickness parameter $d$ for $\alpha =1.0$ (green) and 3.0 (blue). The black dotted lines show the nonrelativistic case for parabolic bands. The strict 2D nonrelativistic limit ($d=0$) is marked by the arrows.

Figure 5

Lowest $s$-type energy eigenvalues of the Dirac-Wannier equation as a function of $\alpha$ with respect to the renormalized single-particle dispersion. The black dashed-dotted line indicates the renormalized band gap. The effective thickness parameter has been set to $d=1.0{\lambda }_C$.

Figure 6

Fine structure of the excitonic spectra, illustrated by the splitting of the lowest $p$-type excitonic states.

Figure 7

Predicted resonance positions for ${\mathrm{MoS}}_2$ on ${\mathrm{SiO}}_2$ as function of effective thickness. The solid lines show the theoretical position of the $1s$ (green), $2s$ (cyan) resonance and the gap (black), the dashed lines mark the target energy value for the A-exciton resonance $E_{1s}$ and the best fit for the thickness parameter.

Figure 8

Calculated spectra for ${\mathrm{MoS}}_2$ within different dielectric environments using the same thickness parameter $d=4.47\AA$. (Top) encapsulated with hBN, using ${\epsilon }_{\mathrm{hBN}}=2.89$, (middle) on ${\mathrm{SiO}}_2$ (${\epsilon }_{{\mathrm{SiO}}_2}=2.1$), (bottom) freely suspended. The red and blue arrows indicate the spectral position of the $A_{2s}$ and $A_{3s}$ exciton resonances, and the black arrows the renormalized gap of the A-exciton series. All spectra have been calculated using a phenomenological dephasing rate $\hslash \gamma =10$ meV.

Figure 9

Calculated spectra different TMDC monolayers supported by a ${\mathrm{SiO}}_2$ substrate. (Top) ${\mathrm{MoSe}}_2$, (middle) ${\mathrm{WSe}}_2$, and (bottom) ${\mathrm{WS}}_2$. The green arrows indicate the spectral position of the $A_{2p}$ exciton resonances, which are several tens of meV below the $A_{2s}$ resonances and are splitted by approximately 10 meV. The spectra have been calculated using an effective thickness parameter of $d=4.96\AA$ for ${\mathrm{MoSe}}_2$, $d=5.17\AA$ for ${\mathrm{WSe}}_2$, and $d=2.78\AA$ for ${\mathrm{WS}}_2$ and a phenomenological dephasing rate $\hslash \gamma =10$ meV.

Figure 10

(Left) Renormalized free-particle transition energies and excitonc positions as function of total layer number $N$ for suspended ${\mathrm{MoS}}_2$ using $d=4.47\AA$. With increasing layer number the number of bands increases accordingly and are classified by the layer index $n=1,\cdots ,N$ (see text for explanation). The black diamonds show the free particle transition energies $E_G(n,n)$ and the orange dots the lowest exciton resonance positions $E_{1s}(n,n)$ in the individual layers. The dashed line indicates the weighted average for the free particle transition, see text for explanation. The dashed-dotted lines indicate the bulk limit $N\to \infty$.

Figure 11

Transition energies for intra- and interlayer excitons in a suspended ${\mathrm{MoS}}_2$ sample consisting of $N=49$ layers. Results are shown for layers with $1\le n\le 25$. Transitions energies for electron-hole pairs where one particle is confined in layers with $n>25$ are identical to those with $n^{\prime }=50-n$ by inversion symmetry. (Left) Free particle transition energy $E_G(n,n)$ (diamonds) and lowest exciton resonance $E_{1s}(n,n)$ (triangles) for electron-hole pairs where both particles are localized within the $n\mathrm{th}$ layer. (Middle) Free particle transition energy $E_G(1,n)$ (diamonds) and lowest exciton resonance $E_{1s}(1,n)$ (triangles) for electron-hole pairs where one particle (here the hole) is localized in the top layer ($n=1$), and the other particle in the $n\mathrm{th}$ layer. (Right) Free-particle transition energy $E_G(25,n)$ (diamonds) and lowest exciton resonance $E_{1s}(25,n)$ (triangles) for electron-hole pairs where one particle (here the hole) is localized in the middle layer ($n=25$), and the other particle in the $n\mathrm{th}$ layer. The $\ominus$ and $\oplus$ symbols in the bottom range of the figures indicate the respective position the electron and hole within the sample and the arrows identify that particle whose position is varied along the figure axis.

Figure 12

Calculated absorption spectra of a suspended mono- and bilayer ${\mathrm{MoS}}_2$. The black lines show the total absorption spectra, whereas the dark blue and dark red fillings correspond to the $A$ and $B$ intralayer contributions, and the light blue and light red fillings to the $A$ and $B$ interlayer contributions, respectively. Spectra have been calculated using a nonradiative homogeneous linewidth $\hslash \gamma =10$ meV.

Figure 13

Imaginary part of the linear susceptibility of ${\mathrm{MoS}}_2$ in the bulk limit using a homogeneous linwidth $\hslash \gamma =10$ meV. The black line shows the imaginary part of the total linear susceptibility, whereas the dark blue filling shows the intralayer contributions, the blue filling the next-neighbor interlayer, and the light blue filling the next-next-nearest-neighbor interlayer excitons contributions respectively. The dashed line around 2.03 eV indicates the position of the renormalized A-exciton gap.

Figure 14

(Left) Nonresonant contributions to the dielectric function for the middle layer of a suspended ${\mathrm{MoS}}_2$ sample consisting of $N$ layers. (Middle) Total effective dielectric function for middle layer of a suspended ${\mathrm{MoS}}_2$ sample with $N$ layers within the strict 2D limit. (Right) Total effective dielectric function for middle layer of a suspended ${\mathrm{MoS}}_2$ sample with $N$ layers including finite size effects with an effective layer thickness $d=4.3\AA$.