Non-Abelian topological defects and strain mapping in 2D moiré materials

Abstract

We present a general method to analyze the topological nature of the domain boundary connectivity that appeared in relaxed moiré superlattice patterns at the interface of 2-dimensional (2D) van der Waals (vdW) materials. At large enough moiré lengths, all moiré systems relax into commensurated 2D domains separated by networks of dislocation lines. The nodes of the 2D dislocation line network can be considered as vortex-like topological defects. We find that a simple analogy to common topological systems with an S 1 order parameter, such as a superconductor or planar ferromagnet, cannot correctly capture the topological nature of these defects. For example, in twisted bilayer graphene, the order parameter space for the relaxed moiré system is homotopy equivalent to a punctured torus. Here, the nodes of the 2D dislocation network can be characterized as elements of the fundamental group of the punctured torus, the free group on two generators, endowing these network nodes with non-Abelian properties. Extending this analysis to consider moiré patterns generated from any relative strain, we find that antivortices occur in the presence of anisotropic heterostrain, such as shear or anisotropic expansion, while arrays of vortices appear under twist or isotropic expansion between vdW materials. Experimentally, utilizing the dark field imaging capability of transmission electron microscopy (TEM), we demonstrate the existence of vortex and antivortex pair formation in a moiré system, caused by competition between different types of heterostrains in the vdW interfaces. We also present a methodology for mapping the underlying heterostrain of a moiré structure from experimental TEM data, which provides a quantitative relation between the various components of heterostrain and vortex-antivortex density in moiré systems.

Anomalous Interfacial Electron-Transfer Kinetics in Twisted Trilayer Graphene Caused by Layer-Specific Localization

“…Our relaxation simulations (SI Figure 2) also show that at θ m ≤ 0.3° the relaxation of these moiré superlattices reestablishes nearly commensurate ABA, BAB, and/or ABC domains with local DOS that should not deviate substantially from those of freestanding ABA and ABC trilayers. This observation is in line with previous experimental ,,, and theoretical studies , of lattice relaxation in bilayer analogues. Therefore, by considering k 0 variations at θ m < 1° in Figure C (which are also within the range of kinetically resolvable k 0 ), we can extract the local rate constant associated with the AAB and AAA stacking domains through eqs and where β i and κ i 0 represent the area fraction and local standard heterogeneous ET rate constant, respectively, for stacking domain i . k M t B 0 = β A A B κ A A B 0 + β A B C κ A B C 0 + β A B A κ A B A 0 + β S P κ S P 0 k A t A 0 = β A A A κ A A A 0 + β A B A κ A B A …”

“…These area fraction distributions after structural relaxation explain the origin of the kinetic modulation observed in Figure C at θ m < 2° as being driven by θ m -dependent area fractions of the “topological defect” , AAB and AAA sites. Our relaxation simulations (SI Figure 2) also show that at θ m ≤ 0.3° the relaxation of these moiré superlattices reestablishes nearly commensurate ABA, BAB, and/or ABC domains with local DOS that should not deviate substantially from those of freestanding ABA and ABC trilayers.…”

Polar meron-antimeron networks in strained and twisted bilayers

“…So far, band topology in the moiré systems has appeared in the electronic structure of magic angle graphene 16,17 , Chern bands in twisted topological insulators 18 , and topological superconductivity in twisted cuprates 19 . The polar meron-antimeron network suggests that moiré materials also exhibit real space topology, echoing recent similar discoveries of topologically nontrivial strain fields in twisted bilayers 20 and magnetic textures in moiré patterned topological insulators 21 .…”

Giant Atomic Swirl in Graphene Bilayers with Biaxial Heterostrain

“…It is thus a classic example of spontaneous symmetry breaking through a pitchfork bifurcation, caused by the applied heterostrain. [35,36] As a final verification we checked that no swirl is obtained in the calculations of a twist moiré (see Figure S7, Supporting Information). All in all, the theoretical results together with the interpretation of the experimental images firmly establish that we have indeed observed a marginally biaxially heterostrained graphene bilayer.…”