Theoretical investigation of 30 degree twisted bilayer graphene*

To understand the unique stability and coupling mechanism in the interesting incommensurate bilayer graphene on top of Pt substrate with 30 degree twisting angles, we performed theoretical derivations and discovered a general interlay scattering mechanism: for any two vectors of one layer, if their difference equals the difference between the reciprocal vectors of the two layers, they can be coupled via a scattering of another layer. This scattering condition is general for any bilayer structures. Based on our first principles calculations, we found that such a coupling mechanism allows a coupling of K and K’ wave vectors of the top layer to couple and form a gap at the M point of the lower layer. This mechanism also enhances a p-d orbital coupling with substrate and explains why the incommensurate bilayer is even more stable than an AB stacking bilayer, where no significant p-d coupling exist. This discovery largely widens the scope of periodic structures and the general coupling condition provides important insights in the design and manufacture of two dimensional heterostructures that may lead to important electronic and spintronic devices.