Glass Patterns in Twisted Disordered Crystals

TL;DR

This work shows that Glass patterns—local, nonperiodic moiré-like features arising from cross-correlated interlayer disorder—are a general consequence of twisting disordered bilayers. It develops a unified framework for Glass patterns and analyzes several models, including correlated Anderson disorder, adatom-driven disorder with lattice relaxation, and magnetic domains in amorphous bilayers. Key results include a mesoscopic mapping between resistivity and the real-space Green's function, a Glass-pattern center acting as an impurity in moiré lattices, LDOS modulations in adatom-based double-stub lattices, and the prediction of central magnetic domains at small twist angles. Together, these findings establish Glass patterns as a generic, experimentally observable feature of twisted disordered systems and provide a blueprint for future theory and experiments.

Abstract

Twisting and stacking two copies of a 2D crystal can produce a long-wavelength periodic interference pattern known as a moiré pattern. Performing the same procedure with an aperiodic structure instead generates a single moiré spot at the rotation center, known as a Glass pattern. We explore the implications of these patterns across a variety of models: they allow measurement of microscopic parameters from mesoscopic resistivity measurements; they generate an impurity that modifies the properties of a moiré lattice at the rotation center; and they allow for domain formation in amorphous magnets. These results establish Glass patterns as a generic feature of twisted disordered systems and provide a framework for future theoretical and experimental exploration.

Figures (12)

• Figure 1: Glass patterns arising from $5\degree$ twist (left) and $1.1\times$ relative scaling (right). For visual efficacy amidrorVolTwo, the dots are located at randomly displaced lattice points rather than completely uncorrelated positions.
• Figure 2: Distorting one copy of an amorphous material by $(x,y)\rightarrow ((1+2t)x,y+t(y^2-1))$ results in two fixed points at $(0,\pm 1)$, each producing Glass pattern. Overlap increases with pattern size as $t$ decreases.
• Figure 3: Twisting two copies of the same disordered crystal produces both a moiré pattern (from the lattice) and a Glass pattern (from disorder correlations). Disorder is represented here by varying point opacities; the left figure is correlated, the right is anticorrelated. In the anticorrelated case, the central moiré cell shows enhanced uniformity, while in the correlated case dot trajectories persist across multiple cells. The relative prominence of these features reflects this visual construction rather than a universal property.
• Figure 4: Construction of the adatom model in \ref{['Sec:adatomrelax']}: (a) Begin with a bilayer crystal with interlayer adatoms. (b) Exfoliate such that the adatoms allocate randomly and independently between layers. (c) Twist or strain one layer. (d) Re-stack: anticorrelation at the center produces a Glass pattern in addition to the moiré modulation.
• Figure 5: Using a 2D version of the model in \ref{['fig:adatom_ex']}, we fix the atomic locations in one layer and allow the other to relax in-plane. Here we indicate the displacement of that layer relative to a rigid twist. Note the changed behavior at the twist center: the Glass pattern inverts the preferred stacking from AB to AA, disrupting the periodic relaxation pattern.