Dislocations in stacking and commensurate-incommensurate phase transition in bilayer graphene and hexagonal boron nitride

Abstract

Dislocations corresponding to a change of stacking in two-dimensional hexagonal bilayers, graphene and boron nitride, and associated with boundaries between commensurate domains are investigated using the two-chain Frenkel-Kontorova model on top of ab initio calculations. Structural transformations of bilayers in which the bottom layer is stretched and the upper one is left to relax freely are considered for gradually increased elongation of the bottom layer. Formation energies of dislocations, dislocation width and orientation of the boundary between commensurate domains are analyzed depending on the magnitude and direction of elongation. The second-order phase transition from the commensurate phase to the incommensurate one with multiple dislocations is predicted to take place at some critical elongation. The order parameter for this transition corresponds to the density of dislocations, which grows continuously upon increasing the elongation of the bottom layer above the critical value. In graphene and metastable boron nitride with the layers aligned in the same direction, where elementary dislocations are partial, this transition, however, is preceded by formation of the first dislocation at the elongation smaller than the critical one. The phase diagrams including this intermediate state are plotted in coordinates of the magnitude and direction of elongation of the bottom layer.

Energetics and Structure of Domain Wall Networks in Minimally Twisted Bilayer Graphene under Strain

“…To describe the energy and structure of domain walls in bilayer graphene analytically, we use the two-chain Frenkel-Kontorova model. 1,28,29,34,35,48 In this model, it is taken into account that both of the layers change their structure to accomodate domain walls. We, nevertheless, assume that the bilayer is supported 2,3,5 and neglect the out-of-plane buckling.…”

“…where coordinate x corresponds to the direction perpendicular to the domain wall, V (u) is the interlayer interaction energy per unit area of the bilayer along the minimum energy path between adjacent AB and BA minima given by eq 2 and K(β) = E cos 2 β + G sin 2 β describes the dependence of the elastic constant on the shear and tensile character of the domain wall. 28,29 Here E = k/(1 − ν 2 ) and G = k/2(1 + ν), where ν is the Poisson's ratio and k is the elastic constant under uniaxial stress (determined by the Young's modulus Y and thickness of graphene layers h as k = Y h). The condition δ∆W/δu = 0 corresponds to the optimal relative displacement u(x) that minimizes the formation energy of the domain wall in eq 3.…”

“…To estimate the contribution of the dislocation nodes, we suppose that the nodes have the shapes of hexagons with the side l D equal to the width of domain walls and given by eq 5 (Figure 3). Since the relative displacement of the layers changes nearly linearly within domain walls, 1,28,29,34,35,48 we can consider the model in which the layers within a dislocation node are uniformly stretched and rotated, i.e. the relative displacement of the layers within the node is described as…”

“…The majority of the previous theoretical works on the structure and energetics of domain walls were devoted to isolated domain walls which do not cross. 1,3,4,[28][29][30][31] The approach developed in these papers allowed to predict the commensurate-incommensurate phase transition 32 related to the formation of the first domain wall in two-dimensional bilayer systems with one layer stretched uniaxially and another layer free, such as bilayer graphene, 1,29 bilayer boron nitride 28,29 and graphene-boron nitride heterostructure. 30 Although it is in principle possible to study the structure of triangular domain wall networks by atomistic 22,25 and multiscale 33 simulations, the number of atoms in the supercell grows rapidly with decreasing the angle of relative rotation of the layers.…”

Commensurate-incommensurate phase transition and a network of domain walls in bilayer graphene with a biaxially stretched layer

“…In stacking dislocations, one of the layers in a bilayer is slightly stretched and the other one slightly compressed and/or there is a different shear strain in the layers so that the stacking changes from the one groundstate stacking to another one. 1,[15][16][17][18] The variation of the stacking is mostly localized in narrow strips with the width much smaller than the size of commensurate domains where the stacking is close to the ground-state one. These narrow strips are referred to as domain walls [2][3][4][5]7,8 or boundaries between commensurate domains.…”