Mapping the twist angle dependence of quasi-Brillouin zones in doubly aligned graphene/BN heterostructures

Abstract

When monolayer graphene is crystallographically aligned to hexagonal boron nitride (BN), a moiré superlattice is formed, producing characteristic satellite Dirac peaks in the electronic band structure. Aligning a second BN layer to graphene creates two coexisting moiré patterns, which can interfere to produce periodic, quasi-periodic or non-periodic superlattices, depending on their relative alignment. Here, we investigate one of the simplest realizations of such a double-moiré structure, graphene encapsulated between two BN layers, using dynamically rotatable van der Waals heterostructures. Our setup allows \textit{in situ} control of the top BN alignment while keeping the bottom BN fixed. By systematically mapping the charge transport as a function of BN angular alignment, we identify the simultaneous signatures of the original moirés, super-moirés, and a third set of features corresponding to quasi-Brillouin zones (qBZ) formed when the system's periodicity becomes ill-defined. Comparing our measurements with theoretical models, we provide the first experimental mapping of the qBZs as a function of angular alignment. Our results establish a direct experimental link between moiré interference and qBZ formation, opening new avenues for engineering electronic structures in multi-aligned 2D heterostructures.

Figures (4)

• Figure 1: Dynamically tunable double-moiré systems.a, Schematics of a dynamically rotatable van der Waals heterostructure with pre-aligned bottom BN. b, Sketch of two coexisting moirés (red and blue parallelograms on each side). In the region where only the two BNs ovelap, a super-moiré lattice is indicated by a purple parallelogram. c, Four-probe resistance as a function of the carrier density for $\theta_{\mathrm{B}}=1.25^{\circ}$ and $\theta_{\mathrm{T}}=30^{\circ}$. The red arrows indicate the satellite peaks from the bottom moiré. Insert: LFM scan of the sample showing the phase channel. Scale bar is 10 . d, In black the four-probe resistance as a function of the carrier density for $\theta_{\mathrm{B}}=1.25^{\circ}$ and $\theta_{\mathrm{T}}=-0.12^{\circ}$. The gray data is plotted for comparison and corresponds to c. The red arrows indicate the satellite peaks from the bottom moiré, blue arrows the top moiré and the purple, green and brown arrows indicate the peaks of the super-moiré. Moiré length for each peak is represented above the arrows. c and d charge transport measurements performed at 10 .
• Figure 2: Experimental results and the super-moiré model.a, Definition of the top and bottom moiré reciprocal lattice (RL) vector as the difference between graphene and top (or bottom) BN's vectors. Due to the difference in size, only a tiny part of the original RL vectors is shown in the zoomed region. Everything else is to scale. b, Equivalent of a Brillouin zone (BZ) corresponding to the moirés' RL vectors. Six vectors are shown, although only two define the BZ. c, The linear combination of the two moiré RL vectors can be used to define the super-moiré lattices. These correspond to the calculated curves in d. d, Colormap of the angle dependence of the carrier density at full filling (four holes per moiré) for the equivalent BZ corresponding to the vectors in c. e, Four-probe resistance as a function of carrier density for different alignments of the top BN while $\theta_{\mathrm{B}}=1.25^{\circ}$, the expected trajectory of super-moiré peaks shown with the different dashed curves. Measurements taken at 10 K. f, Four-probe resistance as a function of carrier density for $\theta_{\mathrm{T}}=0.53$° (black arrow in e). Each of the arrows in f points to a peak in resistance corresponding to the super-moiré model. Only the lilac arrow in this plot has no super-moiré correspondent.
• Figure 3: Quasi-Brillouin zones model.a, Summary of all the mini-gaps observed in the resistance as a function of carrier density for for the different crystallographic alignments of the top BN. Solid lines represent the first five combinations of $(p,q,r,s)$ of the list in c, which also correspond to the super-moiré model. Experimental points that follow these curves have been colored for clarity. All gray points cannot be described only with this combinations. b, Points that cannot be explained in a are now fitted by using different combinations of $(p,q,r,s)$. Symbols with more than one color cannot be attributed to only one parabola. White filled symbols cannot be described by any parabola but given their proximity to another peak are believe to be angle inhomogeneities. Gray symbols cannot be explained by the model, with the used combinations of integers. c, Values of $(p,q,r,s)$ for the original moirés and the quasi-Brillouin zones.
• Figure 4: Effects of the atomic relaxation in double-moiré region. a and b, (top) in plane atomic displacement, $D_{\mathrm{xy}}$, for the same combination of angular alignments $\theta_{\mathrm{T}}=-0.59^{\circ}$ and $\theta_{\mathrm{B}}=1.25^{\circ}$ with different rotation centers. Black hexagonal structure shows the superlattice unit cell. The band structure and respective spectral weight is shown in the bottom of the figure. Full energy range can be seen in Fig. S13. c-f, four probe resistance as a function of the carrier density for the same bottom alignment but four different top alignments. Color bars represent the main superlattice mini-gaps.