Strain solitons and topological defects in bilayer graphene

Abstract

Bilayer graphene has been a subject of intense study in recent years. The interlayer registry between the layers can have dramatic effects on the electronic properties: for example, in the presence of a perpendicular electric field, a band gap appears in the electronic spectrum of so-called Bernal-stacked graphene [Oostinga JB, et al. (2007) Nature Materials 7:151-157]. This band gap is intimately tied to a structural spontaneous symmetry breaking in bilayer graphene, where one of the graphene layers shifts by an atomic spacing with respect to the other. This shift can happen in multiple directions, resulting in multiple stacking domains with soliton-like structural boundaries between them. Theorists have recently proposed that novel electronic states exist at these boundaries [Vaezi A, et al. (2013) arXiv:1301.1690Zhang F, et al. (2013) arXiv:1301], but very little is known about their structural properties. Here we use electron microscopy to measure with nanoscale and atomic resolution the widths, motion, and topological structure of soliton boundaries and related topological defects in bilayer graphene. We find that each soliton consists of an atomic-scale registry shift between the two graphene layers occurring over 6-11 nm. We infer the minimal energy barrier to interlayer translation and observe soliton motion during in situ heating above 1,000°C. The abundance of these structures across a variety of samples, as well as their unusual properties, suggests that they will have substantial effects on the electronic and mechanical properties of bilayer graphene.domain wall | TEM | stacking faults | STEM S pontaneous symmetry breaking, where the ground state of a system has lower symmetry than the underlying Hamiltonian, occurs in systems ranging from magnetism in solids to the Higgs mechanism in high-energy physics. It leads to multiply degenerate ground states, each with a different "broken" symmetry labeled by an order parameter. In the case of a magnet, the spins locally align, creating a magnetization that plays the role of the order parameter. However, the global orientation of the magnetization can be in one of many directions, determined, for example, by the crystal axes. Locally, the system "spontaneously" chooses one such direction based on external constraints or history. Different local regions can have different orientations, and the boundary between adjacent regions is called a domain wall. Mathematically, this boundary takes the form of a soliton that is finite in width but free to move. Other, more complex topological structures are also possible.The stacking of two graphene sheets exhibits analogous physics. Fig. 1A shows the energy of bilayer graphene as a function of the relative in-plane displacement u between the two graphene sheets, which we will use as a continuous-order parameter (1). The energy as a function of u is maximal in the high-symmetry state (u = 0) where one layer is directly on top of the other, called AA stacking (Fig. 1 A, center and B, edges). Away from u = 0 are six ener...

Critical behavior of four-terminal conductance of bilayer graphene domain walls

“…Forming a junction from layer-stacking domain walls requires patterning fewer leads, but as clearly demonstrated in the samples in Ref. 22, the three-fold symmetry of the underlying graphene lattice restricts intersections of these domain walls to be six-terminal structures. Conductance transi-tions in these six-terminal structures can be calculated and analyzed using the framework established in this paper, though the task will be algebraically more intensive.…”

“…In the presence of an interlayer electric field, when the field direction is reversed across a line [15][16][17][18][19][20][21] or when the field is uniform and the layer stacking switches from AB to BA 16,22,23 , the valley-projected Chern number changes by 2 (−2) across the domain wall in valley K (K ). As a result, both types of domain walls host two chiral edge states in each valley with chirality (direction) locked to valley index K or K , as shown in Fig.…”

Structure of graphene and its disorders: a review

“…For AA stacking, each carbon atoms in the second layer directly aligned on the top of another atom in the first layer, while in bilayer graphene with Bernal or AB stacking, a set of atoms in the second layer sit over the empty centers of hexagons in the first layer. As seen in the scanning transmission electron microscopy (STEM) image of AA-stacked bilayer graphene (see Figure 9(c)) [156], all atomic sites are visible in a hexagonal array and show similar brightness. Whereas in Bernal-stacked bilayer, bright spots having hexagonal symmetry and a spacing of 0.25 nm (close to ) are observed in Figure 9(d).…”

“…Hopping integrals t and t’ correspond to the in-plane nearest-neighboring and next-nearest-neighboring hopping respectively, and parameter is associated with the main interlayer hopping in bilayer. Atomic-resolution STEM images of (c) AA- and (d) AB-stacked bilayer graphenes [156]. Hexagonal ring in the first (second) or bottom (top) layer is marked with green (orange) color.…”

Interlayer electron–phonon coupling in WSe2/hBN heterostructures

“…Excitation spectroscopy reveals the double-resonance nature of such enhancement, and identifies the two resonant states to be the A exciton transition of monolayer WSe 2 and a new hybrid state present only in WSe 2 /hBN heterostructures. The observation of an interlayer electron-phonon interaction could open up new ways to engineer electrons and phonons for device applications.Van der Waals heterostructures of atomically thin twodimensional (2D) crystals are a new class of material in which novel quantum phenomena can emerge from layer-layer interactions [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] . For example, electron-electron interactions between adjacent 2D layers can give rise to a variety of fascinating physical behaviours: the interlayer moiré potential between the graphene and hBN layers leads to mini-Dirac cones and the Hofstadter's butterfly pattern in graphene/hBN heterostructures 5-10 ; electronic couplings between MoS 2 and MoS 2 layers lead to a direct-to indirect-bandgap transition in bilayer MoS 2 (refs 11,12); and Coulomb interactions between MoSe 2 and WSe 2 layers lead to interlayer exciton states in MoSe 2 /WSe 2 heterostructures 13,14 .…”