(a) The calculated Berry curvature (red) for the whole valence bands. The black curves and green dots denote the band structures obtained from the DFT calculations and the TB model, respectively. (b) The calculated edge density of states for the armchair edge of the semi-infinite system. (c) and (d) The same as panels (a) and (b) except that the distance (vdW gap) between graphene and the Cr₂Ge₂Te₆ sheet is reduced by 0.3 Å (d = 3.26 Å). (e) The induced exchange splitting (M) in the graphene layer as a function of d − d₀, where d is the vdW distance between graphene and the Cr₂Ge₂Te₆ sheet and d₀ is the corresponding optimized vdW distance (d₀ = 3.56 Å). (f) The SOC induced band gap E_g as a function of d − d₀. Reprinted with permission from Ref. [18]. Copyright 2015 by the American Physical Society.