Intrinsic Layer-Dependent Surface Energy and Exfoliation Energy of van der Waals Materials

Stacking and twisting 2D van der Waals (vdW) layers have become versatile platforms to tune electron correlation. These platforms rely on exfoliating vdW materials down to a single and few vdW layers. We calculate the intrinsic layer-dependent surface and exfoliation energies of typical vdW materials such as, graphite, h-BN, black P, MX$_2$ (M=Mo and W, X=S, Se and Te), MX (M=Ga and In, X=S, Se and Te), Bi$_2$Te$_3$ and MnBi$_2$Te$_4$ using density functional theory. For exchange-correlation functionals with explicit vdW interaction, a single vdW layer always has the smallest surface energy, giving a surface energy reduction when compared to thicker vdW layers. However, the magnitude of this surface energy reduction quickly decreases with increasing number of atomic layers inside the single vdW layer for different vdW materials. Such atomic-layer-dependence in surface energy reduction helps explain the different effectiveness of exfoliation for different vdW materials down to a single vdW layer.