Structural, electronic, vibrational, and elastic properties of graphene/${\mathrm{MoS}}_2$ bilayer heterostructures

Graphene/${\mathrm{MoS}}_2$ van der Waals (vdW) heterostructures have promising technological applications due to their unique properties and functionalities. Many experimental and theoretical research groups across the globe have made outstanding contributions to benchmark the properties of graphene/${\mathrm{MoS}}_2$ heterostructures. Even though some research groups have modeled the graphene/${\mathrm{MoS}}_2$ heterostructures using first-principles calculations, there exist several discrepancies in the results from different theoretical research groups and the experimental findings. In the present work, we revisit this problem by means of first-principles calculations and address the existing discrepancies about the interlayer spacing between graphene and ${\mathrm{MoS}}_2$ monolayers in graphene/${\mathrm{MoS}}_2$ heterostructures and about the location of Dirac points near the Fermi level. We further investigate the electronic, mechanical, and vibrational properties of the optimized graphene/${\mathrm{MoS}}_2$ heterostructures created using $5\times 5/4\times 4$ and $4\times 4/3\times 3$ supercell geometries having different magnitudes of lattice mismatch. The effect of the varying interlayer spacing on the electronic properties of heterostructures is discussed. Our phonon calculations reveal that the interlayer shear and breathing phonon modes, which are very sensitive to the weak vdW interactions, play a vital role in describing the thermal properties of the studied systems. The thermodynamic and elastic properties of heterostructures are further discussed. A systematic comparison between our results and the results reported from other research groups is presented.

Figure 1

(a) and (b) represent the crystal structure of 5:4 graphene/${\mathrm{MoS}}_2$ bilayer heterostructure from two different crystal orientations.

Figure 2

(a) and (b) represent the electronic band structure of 5:4 and 4:3 graphene/${\mathrm{MoS}}_2$ bilayers calculated using DFT-TS + SOC, respectively.

Figure 3

Figure shows the orbital projected electronic band structure (with SOC) of 5:4 graphene/${\mathrm{MoS}}_2$ bilayer heterostructure. Red, green, and orange colors depict projection of C-$p$, Mo-$d$, and S-$p$ orbitals, respectively. (b) Projection of three spin components on the bands near the $K$ point. Here, horizontal axis ranges from $[-0.8,0.8]\times {10}^{-4}{\AA }^{-1}$ with center defined at the $K$ point.

Figure 4

SOC calculated electronic band structure of graphene/${\mathrm{MoS}}_2$ bilayer at 4% compression (red) and 4% expansion (blue) near the Fermi level. Circles depict the bands from C atoms while solid lines depict bands from Mo atoms. The direction of arrows shows the $S_Z$ spin component.

Figure 5

Figure shows change in the location of the Dirac point for different values of interlayer separations in 5:4 graphene/${\mathrm{MoS}}_2$ bilayer. ${\mathrm{\Delta }}_C$ (${\mathrm{\Delta }}_V$) represents the magnitude of energy separation between the Dirac point and the lowest conduction (valence) band of the ${\mathrm{MoS}}_2$ layer, as illustrated in the inset. Green lines depict the bands from ${\mathrm{MoS}}_2$ near the $K$ point. $x$ is the magnitude of the uniaxial strain on the interlayer separation. The positive (negative) values of $x$ refer to expansion (compression).

Figure 6

Isodensity surfaces (green color) at isosurface level $n=0.007$ for graphene/${\mathrm{MoS}}_2$ bilayer heterostructures: (left) 4% interlayer expansion, (center) optimized interlayer separation, and (right) 4% interlayer compression from equilibrium.

Figure 7

Figures (a) and (b) show the calculated phonon dispersion of 5:4 and 4:3 graphene/${\mathrm{MoS}}_2$ bilayer heterostructures, respectively. An enlarged view of the interlayer phonon modes, i.e., two shear modes (red solid lines) and one breathing mode (green solid line), is depicted next to each phonon spectra. Three dotted blue lines represent three acoustic phonon modes ($ZA,TA,LA$). (c) Calculated specific heat. (d) Atom projected phonon density of states (PDOS)for 4:3 heterostructure. A similar spectra with slight variation in frequencies was obtained for 5:4 heterostructure. (e) A schematic illustration of longitudinal shear, transverse shear, and breathing interlayer phonon modes in graphene/${\mathrm{MoS}}_2$ bilayer.