1D Moiré Channels in 2D Materials

• 1D moiré channels are unidirectional periodic potentials in van der Waals heterostructures produced by lattice mismatch, twist, or strain, enabling flat-band physics and quantum-confined states.
• The paper demonstrates that continuum elasticity models accurately capture atomic relaxation in graphene nanoribbon/hexagonal boron nitride systems, predicting periodic domain patterns and stacking slips.
• The findings highlight how engineered 1D moiré channels can be tuned to control gate-dependent quantum confinement and correlated electronic phases in low-dimensional materials.

A one-dimensional (1D) moiré channel is a spatially confined region, typically arising within van der Waals heterostructures or modulated low-dimensional superlattices, where the interplay of lattice mismatch, relative twist, or strain between two atomically thin layers induces a periodic potential that is essentially unidirectional. This moiré modulation leads to new phenomena not present in the constitutive materials, including gate-tunable arrays of quantum-confined states, flat-band physics, and sensitive control over topological and correlated electronic phases. 1D moiré channels can be engineered via various routes, including controlled stacking of nanoribbons on incommensurate substrates, undulation-induced registry modulation in bilayers, twist-angle-induced stripes in non-hexagonal systems, or domain-wall engineering in moiré superlattices.

1. Structural Formation and Continuum Modeling of 1D Moiré Channels

The archetype of a 1D moiré channel is realized by placing a zigzag graphene nanoribbon (GNR) on hexagonal boron nitride (hBN) with a finite twist angle and carefully controlled registry (Okumura et al., 24 Oct 2025). Atomic relaxation in this composite is accurately described within a continuum elasticity theory, where the total energy is the sum of elastic deformation and moiré-induced interlayer binding. In the continuum limit, the in-plane elastic energy for each layer is

$$U_E = \sum_{l=1,2}\!\int d^2r\,\left\{\frac{\lambda^{(l)}+\mu^{(l)}}{2}(u_{xx}^{(l)}+u_{yy}^{(l)})^2 + \frac{\mu^{(l)}}{2}\left[(u_{xx}^{(l)}-u_{yy}^{(l)})^2+4u_{xy}^{(l)\,2}\right]\right\}$$

where $\lambda^{(l)}, \mu^{(l)}$ are Lamé parameters for each material. Minimization of this energy, given the moiré potential from lattice mismatch and/or orientation, yields a periodic domain pattern with alternating commensurate AB′ stacking and domain walls. The resulting atomic-scale relaxation causes the GNR to adopt a locally wavy trajectory, tracing the hBN crystal axes but featuring discrete stacking slips.

Analogous 1D moiré channels can also be produced via undulatory deformation of van der Waals bilayers (e.g., hBN or TMDs) (Li et al., 2024). A sinusoidal height profile along x, [ h(x) = A \sin \Bigl(2\pi x