Cohesive energy in graphene/MoS₂ heterostructures

Abstract

We establish a cohesive law between parallel graphene and MoS₂ sheets based on their van der Waals interaction with Lennard-Jones (LJ) potential. Cohesive energy, force, stress and binding energy per carbon atom are explicitly expressed in terms of parameters in the LJ potential, distance between graphene and each of three layers in MoS₂ sheet, and area density of atoms on each layer. Molecular dynamics simulations are carried out to support analytical results. Analytical results are useful to investigate the interaction between graphene and MoS₂ sheets, and to design nanoelectromechanical systems with graphene/MoS₂ heterostructures.

Appendix

Regarding Eqs. (1) and (3), one gets:

$$\phi_{1} = 4\rho_{C} \rho_{S} \varepsilon_{CS} \int\limits_{0}^{ + \infty } {dx_{u} } \int\limits_{ - \infty }^{{L_{o} }} {\left( {\sigma_{CS}^{12} A - \sigma_{CS}^{6} B} \right)dx_{1} }$$

(28)

where

$$A = \frac{63\pi }{{256\left[ {\left( {x_{1} - x_{u} } \right)^{2} + h_{1}^{2} } \right]^{{{{11} \mathord{\left/ {\vphantom {{11} 2}} \right. \kern-0pt} 2}}} }};\quad B = \frac{3\pi }{{8\left[ {\left( {x_{1} - x_{u} } \right)^{2} + h_{1}^{2} } \right]^{{{5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-0pt} 2}}} }}$$

(29)

It is noted that

$$\int {\frac{{dx_{1} }}{{\left[ {\left( {x_{1} - x_{u} } \right)^{2} + h_{1}^{2} } \right]^{{{{11} \mathord{\left/ {\vphantom {{11} 2}} \right. \kern-0pt} 2}}} }}} = \frac{{\left( {x_{1} - x_{u} } \right)}}{{315h_{1}^{10} r^{*}}}\left[ {128 + 64\left( {\frac{{h_{1} }}{r^{*}}} \right)^{2} + \,48\left( {\frac{{h_{1} }}{r^{*}}} \right)^{4} + 40\left( {\frac{{h_{1} }}{r^{*}}} \right)^{6} + 35\left( {\frac{{h_{1} }}{r^{*}}} \right)^{8} } \right]$$

(30)

where

$${r^{*}} = \sqrt {\left( {x_{1} - x_{u} } \right)^{2} + h_{1}^{2} }$$

(31)

In general, $h_{1} < r^{*}$, by neglecting the higher order term, one can get

$$\int {\frac{{dx_{1} }}{{\left[ {\left( {x_{1} - x_{u} } \right)^{2} + h_{1}^{2} } \right]^{{{{11} \mathord{\left/ {\vphantom {{11} 2}} \right. \kern-0pt} 2}}} }}} \approx \frac{{\left( {x_{1} - x_{u} } \right)}}{{315h_{1}^{10} r^{*}}}\left[ {128 + 64\left( {\frac{{h_{1} }}{r^{*}}} \right)^{2} } \right]$$

(32)

Combining Eqs. (28), (29), (30), and (32) yields

$$\phi_{1} = 4\rho_{C} \rho_{S} \varepsilon_{CS} \int\limits_{0}^{ + \infty } {\left( {\sigma_{CS}^{12} C - \sigma_{CS}^{6} D} \right)dx_{u} }$$

(33)

where

$$C = \frac{\pi }{{20h_{1}^{10} }}\left\{ {2 - \frac{{\left( {x_{u} - L_{o} } \right)h_{1}^{2} }}{{\left[ {\left( {x_{u} - L_{o} } \right)^{2} + h_{1}^{2} } \right]^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} }} - \frac{{2\left( {x_{u} - L_{o} } \right)}}{{\left[ {\left( {x_{u} - L_{o} } \right)^{2} + h_{1}^{2} } \right]^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} }}} \right\} ,$$

(34)

$$D = \frac{\pi }{{8h_{1}^{4} }}\left\{ {2 - \frac{{\left( {x_{u} - L_{o} } \right)\left[ {2\left( {x_{u} - L_{o} } \right)^{2} + 3h_{1}^{2} } \right]}}{{\left[ {\left( {x_{u} - L_{o} } \right)^{2} + h_{1}^{2} } \right]^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} }}} \right\} .$$

(35)

Equation (8) is obtained by integrating Eq. (33) after substituting Eqs. (34) and (35) into it.