Mapping the moiré potential in multi-layer rhombohedral graphene

TL;DR

The paper addresses how the moiré potential from hBN alignment modulates the electronic structure of rhombohedral graphene when coupled to hBN. By combining high resolution STM/STS with continuum-model calculations, the authors image the dispersion of flat bands under gate induced doping and displacement fields, identifying moiré induced band folding and gaps. A key finding is that reproducing site-resolved spectra requires incorporating a moiré potential acting on the top graphene layer with a sign opposite to the bottom layer, highlighting layer-specific moiré effects. The fabrication method enables large area rhombohedral rtG/hBN devices suitable for ultra low temperature exploration of correlated phases and potential proximity-induced spin orbit coupling in heterostructures.

Abstract

Rhombohedral graphene (rG) aligned with hexagonal boron nitride (hBN) has been shown to host flat bands that stabilize various strongly correlated quantum phases, including Mott insulators, integer, and fractional quantum anomalous Hall phases. In this work, we use scanning tunneling microscopy/spectroscopy (STM/STS) to visualize the dispersion of flat bands with doping and applied displacement fields in a hBN-aligned rhombohedral trilayer graphene (rtG)/hBN moiré superlattice. In addition to the intrinsic flat bands of rtG induced by the displacement field, we observe low-energy features originating from moiré potential-induced band folding. Real-space variations of the spectroscopic features allow us to quantify the spatial structure of the moiré potential at the rtG/hBN interface. Importantly, we find that accurately capturing the moiré site-dependent spectra requires incorporating a moiré potential acting on the top graphene layer with a sign opposite to that of the bottom layer into the continuum model. Our results thus provide key experimental and theoretical insights into understanding the role of the moire superlattice in rG/hBN heterostructures.

Figures (5)

• Figure 1: Stabilization and identification of large area trilayer rhombohedral (ABC) graphene. (a) Bernal (ABA) and rhombohedral (ABC) stacking configurations in trilayer graphene. (b) Schematic showing fabrication steps of trilayer ABC graphene aligned with hBN. (c) Raman signal of the fabricated device at room temperature. Inset shows a large area scanning near field optical microscopy (SNOM) map showing uniformity of ABC phase. Scale bar 2 µ m. (d) Measurement schematic of scanning tunneling microscopy/spectroscopy (STM/STS) on trilayer ABC graphene aligned with hBN. Inset: the local stacking configuration within the hBN/graphene moiré of wavelength $\lambda$. Different sites C$_\mathrm{BN}$, C$_\mathrm{B}$, and C$_\mathrm{N}$ of the moiré are labeled. (e) Constant tunneling current (600 mV, 70 pA) topography of the hBN/ABC graphene moiré pattern. Scale bar $30$ nm. Inset: Line cut profile along the arrow showing the inequivalent C$_\mathrm{BN}$, C$_\mathrm{B}$, and C$_\mathrm{N}$ sites. (f) Measured d$I$/d$V$ spectra at site C$_\mathrm{B}$ (marked by red dot in (e)) at $7$ K and at charge neutrality ($V_\text{g}$ = $-32$ V) point. The positions of remote bands are marked by vertical arrows. Inset shows zoomed-in low energy spectra with higher energy resolution showing valence flat band (VFB), conduction flat band (CFB), moiré valence remote band (mVRB), and moiré conduction remote band (mCRB).
• Figure 1: Band structure of ABA and ABC trilayer graphene. (a), (b) Tight-binding band structure of ABA and ABC trilayer graphene, respectively. The solid gray curves are at zero displacement field, the black dashed curves are for a non-zero displacement field that opens up a gap in the system at band crossings. The key difference between ABA and ABC graphene is the energy separation between the higher-energy remote bands. (c) Calculated local density of state (LDOS) of ABA and ABC graphene. Arrows mark the positions of the remote bands. For ABC, the remote bands are separated by $0.8$ eV and for ABA the remote bands are separated by $1.1$ eV.
• Figure 2: Evolution and layer polarization of flat bands with gate voltage in trilayer ABC graphene/hBN moiré. (a) Gate voltage ($V_\text{g}$) dependence of d$I$/d$V$ at site C$_\mathrm{B}$ from $-50$ V to $+50$ V. Four main features in the spectra CFB, VFB, mCRB, and mVRB are marked by red, gray, cyan, and black arrowheads, respectively. Charge neutrality point (CNP) and zero displacement field point ($D = 0$) are identified ($V_\text{g} = -32$ V) with the horizontal arrow. The vertical white dashed line marks the Fermi level. (b) Top panel shows evolution of the energy gap ($\Delta$) between VFB (gray curve in (a)) and CFB (red curve in (a)) with $V_\text{g}$. We marked the $D = 0$ point where $\Delta$ is minimum. Bottom panel shows the evolution of the strength of the CFB, VFB, mCRB, and mVRB peaks with $V_\text{g}$. (c) Layer polarized band structure of the trilayer ABC graphene/hBN moiré at finite electric field values $+0.3$ (top), $0$ (middle), $-0.3$ (bottom) V/nm. Right panels show the corresponding LDOS projected on the top layer. (d) Calculated LDOS at site C$_\mathrm{B}$ projected on the top layer. Calculated spectra share many characteristics with the experimental spectra in (a). (e) Left: schematic for tip-induced doping and displacement field during STM/STS measurements. Right: Accessible parameter space using a single gate sample geometry. Line a traces parameter space expected from a ideal parallel plate capacitor geometry. Line b traces the parameter space used to reproduce (d) that fits the experiment in (a).
• Figure 3: Real space effect of moiré potential as local displacement field. (a)-(c) Moiré potential profiles corresponding to hBN alignments $\xi = \pm 1$, and a moiré potential extracted from topography. The high symmetry moiré stacking sites C$_\mathrm{BN}$, C$_\mathrm{B}$, and C$_\mathrm{N}$ are labeled. The atomistic hBN alignment with ABC graphene for $\xi = \pm 1$ are shown in the insets to (a), and (b). Low-energy sublattice of bottom layer graphene, A1 sits on boron for $\xi = 1$ and on nitrogen for $\xi = -1$. Lower panels show linecuts of the moiré potential $V(r)$ along the high symmetry sites. The orange color denotes positive value of the moiré potential and purple color denote negative value. (d) Under applied $-D$ and in absence of moiré, constant potential landscape in top and bottom graphene. The constant positive potential at bottom layer is denoted by orange line where as the constant negative potential on top layer is denoted by purple line. (e) In presence of moiré on bottom graphene layer the potential landscape modulates with moiré. At some sites the potential due to applied $-D$ adds with the moiré and at other sites they cancel partially. (f) Potential profile on botttom and top graphene layers when an additional negative moiré is imposed on the top graphene layer.
• Figure 4: Theoretical real space LDOS variation for different moiré potentials and comparison with experiment (a)-(c) Calculated LDOS spectra at C$_\mathrm{BN}$ and C$_\mathrm{B}$ sites at $V_g = -50$ V for three different moiré potentials corresponding to hBN alignments $\xi = +1$, $\xi = -1$, and a moiré mimicking topography, respectively. The moiré potentials act only on the bottom layer of rhombohedral trilayer graphene (rTG). All the spectra are normalized by the VFB intensity. (d) Experimentally measured d$I$/d$V$ spectra at C$_\mathrm{BN}$ and C$_\mathrm{B}$ sites for $V_g = -50$. The CFB peaks are marked by red triangles. The measured flat band intensities better match the calculated LDOS spectra in (b) for $\xi = -1$. (e) Calculated LDOS spectra with $\xi = -1$ moiré when a negative ($-40 \%$) $\xi = -1$ moiré potential is additionally imposed onto the top graphene layer. The CFB peaks and moiré induced dips are again marked by red triangles. (f) Calculated C$_\mathrm{BN}$/C$_\mathrm{B}$ CFB peak ratio as a function of the percentage of the $\xi = -1$ moiré potential applied to the top layer. The dashed lines indicate where the theoretical modeling fits to experiment.