Robust structural superlubricity of twisted graphene bilayer and domain walls between commensurate moiré pattern domains from first-principles calculations

Abstract

Twisted graphene layers exhibit extremely low friction for relative sliding. Nevertheless, previous studies suggest that the area contribution to friction for commensurate moiré systems is finite and might restrict macroscopic superlubricity for large layer overlaps. In this paper, we investigate the potential energy surface (PES) for relative displacement of the layers forming moiré patterns (2,1) and (3,1) by accurate density functional theory calculations using the vdW-DF3 functional. The amplitudes of PES corrugations on the order of 0.4 and 0.03 $μ$eV per atom of one layer, respectively, are obtained. The account of structural relaxation doubles this value for the (2,1) pattern, while causing only minimal changes for the (3,1) pattern. We show that different from aligned graphene layers, for moiré patterns, PES minima and maxima can switch their positions upon changing the interlayer distance. The PES shape is closely described by the first spatial Fourier harmonics both with and without account of structural relaxation. A barrier for relative rotation of the layers to an incommensurate state that can make superlubricity robust is estimated based on the approximated PES. We also derive a set of measurable physical properties related to interlayer interaction including shear mode frequency, shear modulus and static friction force. Furthermore, we predict that it should be possible to observe domain walls separating commensurate domains, each comprising a large number of moiré pattern unit cells, and provide estimates of their characteristics.

Figures (6)

• Figure 1: A scheme of the commensurate moiré pattern (2,1) formed by twisted graphene layers. Lattice vectors $\bm{\mathrm{a}}_1$ and $\bm{\mathrm{a}}_2$ of the bottom graphene layer and $\bm{\mathrm{a}}_1'$ and $\bm{\mathrm{a}}_2'$ of the top layer, lattice vectors $\bm{\mathrm{L}}_1$ and $\bm{\mathrm{L}}_2$ of the commensurate moiré pattern, and twist angle $\theta$ corresponding to relative rotation of the graphene layers are indicated.
• Figure 2: Potential energy change $\Delta U$ (in $\mu$eV per atom of one layer) as a function of the relative displacement $r/l$ of the layers along the diagonal of the moiré pattern unit cell for twisted graphene bilayers with commensurate moiré patterns (2,1) (a) and (3,1) (b). The energy is given relative to the PES extrema arranged in a honeycomb lattice. The displacement is given relative to the length $l = L/N_\mathrm{c}$ of the PES lattice vector. For the (2,1) moiré pattern (a), the results for rigid layers at the optimal interlayer distance $d_\mathrm{eq} = 3.4022$ Å (black squares), layers relaxed with constraints on in-plane positions of all atoms (red circles), and those relaxed with only two constrained atoms in the simulation cell (green triangles) are presented. The height of the simulation box is 20 Å. For the (3,1) moiré pattern (b), the results for rigid layers at the optimal interlayer distance $d_\mathrm{eq} = 3.4003$ Å obtained for the height of the simulation box of 30 Å (black squares) are shown. Solid lines correspond to the approximation by the first Fourier harmonics according to Eq. (\ref{['eq_approx']}).
• Figure 3: Potential energy change $\Delta U$ (in $\mu$eV per atom of one layer) as a function of the relative displacement $r/l$ of the layers along the diagonal of the moiré pattern unit cell for twisted graphene bilayer with the (3,1) moiré pattern. The energy is given relative to the PES extrema arranged in a honeycomb lattice. The displacement is given relative to the length $l = L/N_\mathrm{c}$ of the PES lattice vector. The results for rigid layers at the optimal interlayer distance $d_\mathrm{eq} = 3.4003$ Å (black squares), layers relaxed with constraints on in-plane positions of all atoms (red circles), and those relaxed with only two constrained atoms in the simulation cell (green triangles) are presented. The results are obtained for the height of the simulation box of 20 Å.
• Figure 4: Energy difference $\Delta U_\mathrm{ex}$ between the extrema of the potential energy surface located in vertices of the triangular and honeycomb lattices (in $\mu$eV per atom of one layer, $|\Delta U_\mathrm{ex}| = \Delta U_\mathrm{max}$) for the (2,1) commensurate moiré pattern as a function of the interlayer distance $d$ (in Å) computed using the vdW-DF3 (black line) and PBE (red dashed line) exchange-correlation functionals. The PES types corresponding to different signs of $\Delta U_\mathrm{ex}$ are indicated.
• Figure 5: Potential energy change $\Delta U$ (in $\mu$eV per atom of one layer) as a function of the relative displacement $r/l$ of the layers along the diagonal of the moiré pattern unit cell for the (2,1) commensurate moiré pattern computed using the vdW-DF3 (black line) and PBE (red dashed line) exchange-correlation functionals at the interlayer distance $d=3.34$ Å. The energy is given relative to the PES minimum. The displacement is given relative to the length $l = L/N_\mathrm{c}$ of the PES lattice vector.