Fig. 1.

The energy landscape for interlayer translation in bilayer graphene leads to certain allowed boundary transitions. (A) van der Waals energy landscape for translating one graphene layer across another (data from ref. 1) with the corresponding orientations of the two layers shown schematically in orange and teal. The central location corresponds to AA stacking, having an order parameter vector u = 0. Around this are six energy minima where |u| = a, corresponding to Bernal-stacked graphene. The two mirror-symmetric phases of Bernal-stacked graphene, AB and BA, are related to each other by three distinct low-energy translation directions |Δu| = a indicated by red, green, and blue arrows. (B) A horizontal line cut through the energy landscape in B, along an armchair direction, reveals that AB is connected to BA through a saddle point (SP) having an energy of 2.1 meV/atom, a factor of 10 lower than the energy of AA-stacked graphene. Across this cut, from left to right, the upper graphene sheet, shown in orange, translates to the right with respect to the lower sheet, shown in teal. (C and D) Dark-field TEM images of bilayer graphene, imaged through an aperture in the diffraction plane, as indicated by circles in the inset. The bilayer graphene is supported by approximately two additional graphene sheets at 16° and 31° relative to the bilayer, which are invisible when imaging although the selected diffraction angles. (C) At nonzero sample tilt, selecting electrons from the [−1010] family of diffraction angles enables us to distinguish AB (gray) from BA (black) domains. (D) Three DF-TEM images taken from the [−2110] diffraction angles indicated in the inset are overlaid in red, blue, and green. Imaged this way, each line is an AB–BA domain boundary, with its color indicating the armchair direction along which the relative translation between graphene layers occurs.