One-dimensional moiré engineering in zigzag graphene nanoribbons on hBN

TL;DR

This work addresses how structural relaxation in a one-dimensional moiré system formed by a zigzag graphene nanoribbon on hBN shapes its electronic structure. It develops an effective grid model based on continuum elasticity to compute relaxed geometries across twist angles $\theta$ and ribbon widths, followed by a tight-binding description that includes hBN to compute band structures and local density of states in a 1D moiré cell. The results show that relaxation creates AB' domains separated by alpha and beta domain walls, and the moiré potential strongly modulates zero-energy zigzag edge states, producing dense subbands within AB' domains and sharply localized domain-wall states; gate-tuning can relocate the localization between domain centers and walls, enabling a programmable 1D chain of quantum-confined states. This work establishes 1D moiré engineering in GNR/hBN as a versatile platform for edge-state control and nanodevice design, with potential for Coulomb blockade phenomena and gate-controlled transport in 1D moiré nanostructures.

Abstract

We study the structural relaxation and electronic properties of a one-dimensional (1D) moiré system composed of a zigzag graphene nanoribbon (GNR) placed on a hexagonal boron nitride (hBN) substrate. Using an effective grid model derived from continuum elasticity theory, we calculate the relaxed atomic structure of the GNR/hBN system for various twist angles and ribbon widths. The relaxation gives rise to a characteristic 1D domain structure consisting of alternating commensurate AB$'$ regions and two distinct types of domain boundaries. At finite twist angles, the ribbon adopts a wavy shape, locally tracing the hBN zigzag direction but occasionally sliding to adjacent atomic rows. The resulting moiré potential strongly modulates the electronic structure: the zero-energy zigzag edge states are modulated by the local stacking, leading to densely packed subbands in the AB$'$ domains and sharply localized domain-wall states in the energy gaps between domain plateaus, which together realize gate-tunable one-dimensional arrays of quantum-confined electronic states. Our results demonstrate that moiré modulation in GNR/hBN heterostructures provides a versatile platform for electronic structure engineering and the design of 1D moiré nanodevices.

Figures (8)

• Figure 1: Schematic of the graphene/hBN moiré superlattice. (a) Atomic configuration. (b) Local stacking arrangements: AA$'$, AB$'$, and BA$'$. (c) Lattice structure and moiré pattern at $\theta = 4.08^\circ$ with $(m,n)=(-1,4)$. The horizontal lines indicate the GNR region with $N=10$. (d) Contour plot of the interlayer binding energy corresponding to panel (c). The one-dimensional superlattice period is given by a linear combination of the moiré lattice vectors, $\mathbf{\Lambda} = m \mathbf{L}^{\mathrm{M}}_1 + n \mathbf{L}^{\mathrm{M}}_2$, represented by the white arrows.
• Figure 2: Moiré patterns for different twist angles, similar to Fig. \ref{['g_hbn_4.08deg']}(d). In each panel, the green hexagon represents a unit cell of the 2D moiré pattern, spanned by the moiré lattice vectors $\mathbf{L}^\mathrm{M}_1$ (red arrow) and $\mathbf{L}^\mathrm{M}_2$ (blue arrow). The horizontal lines indicate the GNR region with $N = 10$.
• Figure 3: (a) Discrete square-grid model. Graphene and hBN layers are represented by parallel square grids with spacing $l_0$ and identical orientation, independent of the twist angle. Grid vertices correspond to mass points, and springs connect them along orthogonal and diagonal directions. (b) A single grid cell of the spring–mass model and its deformation (see text).
• Figure 4: Optimized structures of zigzag GNRs with width $N=10$ on hBN at twist angles (a) $\theta_{1,0}=0^{\circ}$, (b) $\theta_{2,1}=0.35^{\circ}$, (c) $\theta_{2,3}=0.77^{\circ}$, (d) $\theta_{0,1}=1.86^{\circ}$, (e) $\theta_{-1,6}=2.91^{\circ}$, and (f) $\theta_{-1,4}=4.08^{\circ}$. In each panel, the top and middle plots show the interlayer binding energies before and after relaxation, respectively, while the bottom panel shows the corresponding relaxed lattice structure with the vertical axis magnified by a factor of four. The symbols $\alpha$ and $\beta$ denote the two types of domain walls (see text).
• Figure 5: Interlayer binding energy of GNR on hBN, similar to Fig. \ref{['optimized_energy_structure']}, at twist angles (a) $\theta_{1,0}=0^{\circ}$, (b) $\theta_{2,3}=0.77^{\circ}$, and (c) $\theta_{-1,6}=2.91^{\circ}$, for ribbon widths $N=10, 20, 40,$ and $60$. In each panel, the upper and lower plots correspond to the non-relaxed and relaxed structures, respectively.