Imaging supermoiré relaxation in helical trilayer graphene

Abstract

In twisted van der Waals materials, local atomic relaxation can alter the underlying electronic structure. Characterizing lattice reconstruction and its susceptibility to strain is essential for understanding emergent electronic states, especially in multilayers in which interference between moiré lattices yields larger supermoiré patterns whose energy is highly sensitive to local stacking. Here we image spatial modulations in the electronic character of helical trilayer graphene, which indicate relaxation into a superstructure of large domains with uniform moiré periodicity. We show that the supermoiré domain size is increased by strain and can be altered in the same device while preserving the local properties within each domain. Finally, we observe a higher conductance at the domain boundaries, consistent with predictions that they host counter-propagating edge modes. Our work provides a real-space visualization of moiré-periodic domains, reveals two independently tunable length scales and demonstrates strain engineering as a route towards designing correlated topological networks at the supermoiré scale.

Figures (14)

• Figure 1: Helical trilayer graphene (HTG) electronic structure. a, Left: schematic of the measurement setup (see Methods). The HTG sample consists of three graphene layers consecutively twisted by $\theta \approx 1.5^\circ$ and is encapsulated by hexagonal boron nitride (hBN; top hBN not shown) with a graphite bottom gate. Right: schematic of the supermoiré domains predicted in HTG. The lattice relaxes into triangular domains (denoted as $h$ and $\bar{h}$) with uniform moiré periodicity, which are separated by domain walls (black lines) that intersect at AAA stacking sites (white circles). b, Optical micrograph of the HTG device. Scale bar (white): 2 $\mu$m. c, Inverse electronic compressibility d$\mu$/d$n$ (red) and chemical potential $\mu$ (blue) as a function of carrier density $n$ at temperature $T = 330$ mK. d, HTG band structure in a domain center for $\theta = 1.45^\circ$ and at displacement field $D=0$. Inset: the moiré Brillouin zone with the plotted path in momentum space indicated by red lines and arrows. e-f, Spatial line cut of d$\mu$/d$n$ as a function of $n$ at $T = 1.6$ K (e) and corresponding local twist angle (f) along the black line in b.
• Figure 2: Imaging supermoiré domains. a, Spatial map of $\theta$ within the white box in Fig. \ref{['fig:htg']}b. Scale bar: 500 nm. b, Value of d$\mu$/d$n$ at moiré filling factor $\nu = 4$ in the same area as in a. c, The local minima in b (indicated by black dots) correspond to the AAA stacking sites of HTG and form a triangular lattice with supermoiré wavelength $\lambda_\mathrm{SM}$. The color of each triangular supermoiré domain indicates the ratio of its experimentally observed area $A_\mathrm{exp}$ to the theoretically predicted area $A_\mathrm{th}$ in the absence of strain and for equal interlayer angles. d-f, Same as a-c, but in the red box in Fig. \ref{['fig:htg']}b in a subsequent round of measurements after thermal cycling and other device changes. g, Dependence of $\lambda_\mathrm{SM}$ on the twist angle mismatch $\delta \theta$, where the two interlayer angles are $\theta \pm \delta \theta/2$ and $\theta = 1.45^\circ$. h, $\lambda_\mathrm{SM}$ as a function of global isotropic biaxial heterostrain $\epsilon$ on the middle layer for $\theta = 1.45^\circ$ and $\delta \theta = 0$. The pink shaded area denotes the narrow range for which $\lambda_\mathrm{SM}$ exceeds its predicted value in the absence of strain (black dashed lined). i, Histogram of the observed $A_\mathrm{exp}$ before and after thermal cycling. Black dashed line indicates $A_\mathrm{th}$ for $\theta = 1.45^\circ$ assuming $\epsilon=0$ and $\delta \theta = 0$. The "After" histogram includes domains outside the field of view of panel f (Extended Data Fig. 6). All data measured at $T = 1.6$ K.
• Figure 3: Enhanced domain wall conductance. a-b, High-resolution spatial maps of the change in chemical potential $\Delta \mu_{4}$ at $\nu = 4$ measured with a.c. (a) and d.c. (b) modalities. Scale bar: 500 nm. c, $\mu$ as a function of $n$ in the vicinity of $\nu = 4$, measured using a.c. (red) and d.c (blue) modalities. d, $\Delta \mu_{4}$ for a.c. (red) and d.c. (blue) modalities along the black trajectory from bottom left to top right in b. e, Schematic showing the predicted counter-propagating topological edge modes (for one spin) at $\nu=4$ along the boundaries of the $h$ and $\overline{h}$ domains in HTG. f-g, Band structure of HTG at a domain wall (f) and AAA site (g) at $D=0$. h, Theoretical spatial dependence of $\Delta {\mu}_{4}$ based on band structure calculations within the domains, at the domain walls, and at the AAA sites for $\theta=1.45^\circ$ and $D=0.45$ V/nm (see Supplementary Sec. 1c), with the domain size enlarged to match the experiment. Scale bar: 500 nm. i, Simulated spatial dependence of $\Delta {\mu}_{4}$ after accounting for the finite resolution of the SET tip. We assume a tip height of $h = 70$ nm and radius $R = 70$ nm (Supplementary Sec. 5). All data measured at $T = 1.6$ K.
• Figure 4: Magnetic field dependence.a, d$\mu$/d$n$ as a function of $\nu$ and perpendicular magnetic field $B$ in the center of a triangular domain at $T = 330$ mK. b, Wannier diagram of incompressible states identified from a. Black, red, orange, blue and grey states respectively correspond to states with integer zero-field intercepts at $|\nu| = 0, 1, 2, 3$, and $4$. c, Calculated Wannier-like diagram showing Hofstadter gaps for HTG at $\theta = 1.45^{\circ}$. The size of each dot reflects the gap magnitude.
• Figure ED1: Moiré-periodic order in HTG. Schematic of the local stacking configurations within $h$ and $\bar{h}$ domains and AAA sites. The domains exhibit moiré-periodic order: a honeycomb lattice of alternating AAB and BAA stackings (orange and purple, respectively) with uniform moiré wavelength $\lambda_{\rm{M}}$, surrounded by local ABA stacking (white). The $h$ and $\bar{h}$ domains are related by a $C_{2z}$ transformation. The outermost schematics show the corresponding local real-space alignment of the bottom, middle, and top graphene layers in red, green, and blue, respectively. Here, A and B refer to the sublattices of the monolayer graphene honeycomb lattices.