^{1,a)}, Bing-Shen Wang

^{2}and Timon Rabczuk

^{1,3,a)}

### Abstract

The interlayer energy of the twisting bilayer graphene is investigated by the molecular mechanics method using both the registry-dependent potential and the Lennard-Jones potential. Both potentials show that the interlayer energy is independent of the twisting angle θ, except in the two boundary regions ° or 60∘, where the interlayer energy is proportional to the square of the twisting arc length. The calculation results are successfully interpreted by a single atom model. An important information from our findings is that, from the energy point of view, there is no preference for the twisting angle in the experimental bilayer graphene samples, which actually explains the diverse twisting angles in the experiment.

We would like to thank the innominate referee for suggesting the registry-dependent interlayer potential. The work was supported in part by the Grant Research Foundation (DFG).

I. INTRODUCTION

II. RESULTS AND DISCUSSION FOR REGISTRY-DEPENDENT POTENTIAL

III. RESULTS AND DISCUSSION FOR LENNARD-JONES POTENTIAL

IV. CONCLUSION

### Key Topics

- Graphene
- 18.0
- Carbon
- 8.0
- Density functional theory
- 6.0
- Atomic electronic properties
- 4.0
- Mechanical properties
- 4.0

## Figures

The interlayer potential energy calculated from the registry-dependent potential with different cut-offs. The y-axis (Ea ) is the total interlayer energy per atom. In this calculation, the radius of the top graphene layer is R = 100 Å. The twisting angle °, i.e., AB-stacking BLG. For a cut-off , the variation in the energy is on the order of 10−3 meV. Inset displays the cross section of a twisting BLG, where R is the radius and rc is the boundary region.

The interlayer potential energy calculated from the registry-dependent potential with different cut-offs. The y-axis (Ea ) is the total interlayer energy per atom. In this calculation, the radius of the top graphene layer is R = 100 Å. The twisting angle °, i.e., AB-stacking BLG. For a cut-off , the variation in the energy is on the order of 10−3 meV. Inset displays the cross section of a twisting BLG, where R is the radius and rc is the boundary region.

The saturation of the energy per atom with increasing radius for different twisting angles θ. (a) °, i.e., AB-stacking BLG. Solid and open dots correspond to cut-off and 50 Å. (b) , i.e., AA-stacking BLG. Theenergy per atom in the AA-stacking BLG is radius independent. (c) , i.e., the first commensurable angle. (d) , i.e., the second commensurable angle.

The saturation of the energy per atom with increasing radius for different twisting angles θ. (a) °, i.e., AB-stacking BLG. Solid and open dots correspond to cut-off and 50 Å. (b) , i.e., AA-stacking BLG. Theenergy per atom in the AA-stacking BLG is radius independent. (c) , i.e., the first commensurable angle. (d) , i.e., the second commensurable angle.

The energy per atom versus twisting angle θ. The whole curve is symmetric about due to the mirror symmetry between the two nonequivalent atoms in graphene, so is not shown. Inset shows the interlayer space verses twisting angle θ.

The energy per atom versus twisting angle θ. The whole curve is symmetric about due to the mirror symmetry between the two nonequivalent atoms in graphene, so is not shown. Inset shows the interlayer space verses twisting angle θ.

The energy per atom versus twisting angle around °. For curves from bottom to top, the radius increases from 10 to 100 Å. All curves can be well fitted to , where is the value at °. α is the fitting parameter. Inset shows the radius-dependence of α. α is fitted to .

The energy per atom versus twisting angle around °. For curves from bottom to top, the radius increases from 10 to 100 Å. All curves can be well fitted to , where is the value at °. α is the fitting parameter. Inset shows the radius-dependence of α. α is fitted to .

The energy per atom versus twisting angle around . For curves from top to bottom, the radius increases from 10 to 100 Å. All curves can be well fitted to , where is the value at . β is the fitting parameter. Inset shows the radius-dependence of β. β is fitted to .

The energy per atom versus twisting angle around . For curves from top to bottom, the radius increases from 10 to 100 Å. All curves can be well fitted to , where is the value at . β is the fitting parameter. Inset shows the radius-dependence of β. β is fitted to .

The registry-dependent potential between a single atom and an infinite large graphene sheet. The single atom is on top of the graphene at a fixed distance z = 3.478 Å. x and y axes in the figure are the other two coordinates of the single atom. (a) shows six-fold symmetry in the energy, due to the hexagon structure of graphene. (b) shows only one valley of the energy. (c) shows the convergence of the average of the energy over xy area with increasing grid points. The dashed line (red) depicts the platform value of the energy per atom in Fig. 3 .

The registry-dependent potential between a single atom and an infinite large graphene sheet. The single atom is on top of the graphene at a fixed distance z = 3.478 Å. x and y axes in the figure are the other two coordinates of the single atom. (a) shows six-fold symmetry in the energy, due to the hexagon structure of graphene. (b) shows only one valley of the energy. (c) shows the convergence of the average of the energy over xy area with increasing grid points. The dashed line (red) depicts the platform value of the energy per atom in Fig. 3 .

The interlayer potential energy calculated from Lennard-Jones potential with different cut-offs. The y-axis (Ea ) is the energy per atom. In this calculation, the radius of the top layer is 100 Å. The twisting angle . For a cut-off distance of , the variation in the energy is on the order of 10−5 meV.

The interlayer potential energy calculated from Lennard-Jones potential with different cut-offs. The y-axis (Ea ) is the energy per atom. In this calculation, the radius of the top layer is 100 Å. The twisting angle . For a cut-off distance of , the variation in the energy is on the order of 10−5 meV.

The saturation of the energy per atom with increasing radius for different twisting angles θ. (a) °, i.e., AB-stacking BLG. (b) , i.e., AA-stacking BLG. The energy per atom in the AA-stacking BLG is radius independent. (c) , i.e., the first commensurable angle. (d) . (e) , i.e., the second commensurable angle. (f) .

The saturation of the energy per atom with increasing radius for different twisting angles θ. (a) °, i.e., AB-stacking BLG. (b) , i.e., AA-stacking BLG. The energy per atom in the AA-stacking BLG is radius independent. (c) , i.e., the first commensurable angle. (d) . (e) , i.e., the second commensurable angle. (f) .

The energy per atom versus twisting angle θ. The top inset shows the close-up of the middle region , where curves for large radius of 500 and 1000 Å are indistinguishable. Two bottom insets show the close-up of the boundary regions ° and 60°.

The energy per atom versus twisting angle θ. The top inset shows the close-up of the middle region , where curves for large radius of 500 and 1000 Å are indistinguishable. Two bottom insets show the close-up of the boundary regions ° and 60°.

The energy per atom versus twisting angle around °. For curves from bottom to top, the radius increases as 20, 55, 150, 190, 250, 310, 400, 520, 665, 850, and 1000 Å. All curves can be well fitted to , where is the value at °. α is the fitting parameter. Inset shows the radius-dependence of α. α is fitted to .

The energy per atom versus twisting angle around °. For curves from bottom to top, the radius increases as 20, 55, 150, 190, 250, 310, 400, 520, 665, 850, and 1000 Å. All curves can be well fitted to , where is the value at °. α is the fitting parameter. Inset shows the radius-dependence of α. α is fitted to .

The energy per atom versus twisting angle around . For curves from top to bottom, the radius increases as 20, 55, 150, 190, 250, 310, 400, 520, 665, 850, and 1000 Å. All curves can be well fitted to , where is the value at . β is the fitting parameter. Inset shows the radius-dependence of β. β is fitted to .

The energy per atom versus twisting angle around . For curves from top to bottom, the radius increases as 20, 55, 150, 190, 250, 310, 400, 520, 665, 850, and 1000 Å. All curves can be well fitted to , where is the value at . β is the fitting parameter. Inset shows the radius-dependence of β. β is fitted to .

The Lennard-Jones potential between a single atom and an infinite large graphene sheet. The single atom is on top of the graphene at a fixed distance z = c = 3.35 Å. (a) shows six-fold symmetry in the energy, due to the hexagon structure of graphene. (b) shows only one valley of the energy. (c) shows the convergence of the average of the energy over xy area with increasing grid points. The dashed line (red) depicts the platform value of the energy per atom in Fig. 9 .

The Lennard-Jones potential between a single atom and an infinite large graphene sheet. The single atom is on top of the graphene at a fixed distance z = c = 3.35 Å. (a) shows six-fold symmetry in the energy, due to the hexagon structure of graphene. (b) shows only one valley of the energy. (c) shows the convergence of the average of the energy over xy area with increasing grid points. The dashed line (red) depicts the platform value of the energy per atom in Fig. 9 .

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