Band gap formation in commensurate twisted bilayer graphene/hBN moiré lattices

We report on the investigation of periodic superstructures in twisted bilayer graphene (tBLG) van der Waals heterostructures, where one of the graphene layers is aligned to hexagonal boron nitride (hBN). Our theoretical simulations reveal that if the ratio of the resulting two moiré unit-cell areas is a simple fraction, the graphene/hBN moiré lattice acts as a staggered potential, breaking the degeneracy between tBLG AA sites. This leads to additional band gaps at energies where a subset of tBLG AA sites is fully occupied. These gaps manifest as Landau fans in magnetotransport, which we experimentally observe in an aligned tBLG/hBN heterostructure. Our study demonstrates the identification of commensurate tBLG/hBN van der Waals heterostructures by magnetotransport, highlights the persistence of moiré effects on length scales of tens of nanometers, and represents an interesting step forward in the ongoing effort to realize designed quantum materials with tailored properties.

Figure 1

(a) Schematic representation of a tBLG moiré lattice. Gray circles represent the AA sites while the gray rhombus denotes the tBLG unit cell. (b) Schematic representation of a composite supermoiré lattice consisting of a tBLG moiré lattice [same as in (a)] and a graphene/hBN (gr/hBN) moiré lattice. Blue circles represent the AA sites of the resulting supermoiré lattice. The shaded rhombus denotes the corresponding supermoiré unit cell.

Figure 2

(a) Band structure and density of states calculation of a tBLG moiré system at a twist angle of ${\theta }_{\mathrm{tBLG}}=0.{987}^{\circ }$ (black arrows mark the tBLG moiré-induced single-particle band gaps). To distinguish bulk from edge states in our ribbon geometry (width $W\approx 210$ nm), we color each line according to the probability $P_{{\left|\mathrm{\Psi }\right|}^2}$ to find the Bloch state at the center of the ribbon, i.e., within $[0.2,0.8]\times W$. (b) Spatial distribution of the charge carrier density in one graphene layer, averaged over different energy windows (shaded areas). The charge carrier density accumulates at the AA sites located at the corners of the tBLG moiré unit cell (see sketched rhombus) forming a uniform triangular lattice in real space. One tBLG moiré unit cell contains one AA site (proportional contributions from the corners). The $y$ axis is identical for all panels. (c), (d) Same as in (a) and (b) but with an additional hBN alignment of ${\theta }_{\mathrm{gr}/\mathrm{hBN}}=0.{62}^{\circ }$. Blue arrows mark the positions of the additional gaps forming due to the hBN alignment. Black rhombus in the upmost panel represents the unit cell of the supermoiré lattice defined by the tBLG and the graphene/hBN moiré lattices. The supermoiré unit cell contains a total of 3 AA sites. On this scale three tBLG moiré unit cells feature the same area as four graphene/hBN moiré unit cells (second panel from top). The presence of the graphene/hBN moiré strongly modulates the electron density on the different AA sites of the tBLG moiré lattice.

Figure 3

(a) Line cuts of the charge carrier density distribution averaged over specific energy ranges as shown in Fig. 2 for $y=25\mathrm{nm}$ of the tBLG system. The energy ranges are given by $E_1\in [2.5,12]\mathrm{meV}, E_2\in [1.3,2.5]\mathrm{meV}, E_3\in [-0.153,1.3]\mathrm{meV},$ and $E_4\in [-4.2,-0.153]\mathrm{meV}$. The charge distribution on the different AA sites is uniform for tBLG. The $y$ axis is identical for all panels. (b) Same as in (a) but with the additional hBN alignment of ${\theta }_{\mathrm{gr}/\mathrm{hBN}}=0.{62}^{\circ }$ [line cuts are taken from corresponding panels in Fig. 2]. The energy ranges are given by $E_1\in [11,7]\mathrm{meV}, E_2\in [-0.4,7]\mathrm{meV}, E_3\in [-6,-0.4]\mathrm{meV}$, and $E_4\in [-11,-6]\mathrm{meV}$. The presence of the graphene/hBN moiré lattice strongly modulates the electron density on the tBLG AA sites.

Figure 4

(a) Magnetoresistance $R$ [see Eq. (1)] simulation of a tBLG system with a twist angle of ${\theta }_{\mathrm{tBLG}}=0.{987}^{\circ }$. Landau levels emerge from the charge neutrality point $(\nu =0)$ and the single-particle band gaps at full filling of the tBLG moiré lattice unit cell $(\nu =4)$. (b) Same as in (a) but for a composite tBLG/hBN system with an additional hBN alignment of ${\theta }_{\mathrm{gr}/\mathrm{hBN}}=0.{62}^{\circ }$. For this case there are additional Landau levels emerging from noninteger filling factors of $({\nu }_{\mathrm{sat}}\approx \pm 2.67)$ (see black arrows).

Figure 5

(a) Magnetic field derivative of the longitudinal resistance as a function of the filling factor $\nu$ and normalized magnetic flux per tBLG moiré superlattice unit cell $\varphi$. We observe clear Landau levels emerging from charge neutrality and an integer filling factor of $\nu =2$. The Landau levels emerging from $\nu =\pm 4$ are less pronounced. Furthermore, additional Landau levels emerging from a noninteger filling factor of around $\nu \approx \pm 2.5\equiv {\nu }_{\mathrm{sat}}$ tilted towards the charge neutrality point are visible. (b) Zoom-in into the relevant area from (a). The lines are plotted as a guide to the eye to highlight relevant Landau fans at the charge neutrality point (light blue), the integer filling $\nu =+2$ (dark blue), and at the noninteger filling factor $\nu \approx \pm 2.5$ (black).

Figure 6

Optical image of the fabricated van der Waals heterostructure. The dashed lines show the outlines of graphite (black), tBLG (green), top hBN (red), and bottom hBN (blue) flakes. The graphene/hBN moiré lattice is formed on the interface between the bottom hBN and a single-layer graphene flake. Solid lines show the crystallographic axes used for the (mis)alignment during the fabrication. The angles are given with respect to the horizontal white line.

Figure 7

(a) Atomic force microscopy image of the measured device. The electrical connections for the two- and four-terminal measurements are depicted. (b) Schematic cross section of the device showing the composition of the heterostructure. (c) Two-terminal differential conductance $\mathrm{d}I/\mathrm{d}{V}_{2\mathrm{T}}$ as a function of the charge carrier density $n$ for different temperatures measured along the entire Hall bar structure. The minima in conductance correspond to the band insulators (BI), correlated insulators (CI), and the charge-neutrality point (CNP) (d) Arrhenius plot of the differential resistance $\mathrm{d}V/\mathrm{d}{I}_{2\mathrm{T}}$ of the band-insulating states as a function of the inverse temperature. The individual data points are extracted by taking the mean of the minima in the band-insulating states. The error bars represent the standard deviations. (e) Same as in (d), but for the correlated-insulating states.

Figure 8

(a) Two-terminal differential resistance as a function of magnetic field and carrier density measured along the entire Hall bar structure [compare with Fig. 7]. (b) Four-terminal differential resistance measured in the configuration shown in Fig. 7 as a function of magnetic field and carrier density. The black arrows mark the positions where the Landau levels emerging from the graphene/hBN moiré lattice intersect with the Landau levels emerging from the tBLG moiré lattice. Visible is the narrowing of the lowest Landau level emerging from the charge-neutrality point at the certain intersection positions.

Figure 9

(a) Finite-bias spectroscopy on the hole-doped BI state showing individual Coulomb diamonds. (b) Same as in (a) but for the electron-doped BI. (c) Schematic illustration of the measurement setup.

Figure 10

(a) Closeup of twisted bilayer graphene (large angle for clarity) (black: top layer, gray: bottom layer). (b) Unit cell in the $x-y$ plane with local displacement vector $\vec{d}$. (c) Unit cell of tBLG. Each local configuration can be mapped to a unit cell (see insets) with a corresponding shift.

Figure 11

(a) Schematic of the real-space moiré superlattice unit cell of the graphene/hBN system. Colored circles (I,II,V) correspond to centers of Gaussians used in the effective moiré potential. (b) Schematic explanation of the superposition of the two moiré lattices. Light gray lines indicate edge character of the unit cells. Assuming a small graphene/hBN twist angle of ${\theta }_{\mathrm{gr}/\mathrm{hBN}}\approx 0.{62}^{\circ }$ brings the two moiré cells to the same periodicity. (c) Lattice constant of the graphene/hBN moiré lattice as a function of their relative twist angle ${\theta }_{\text{gr/hBN}}$.

Figure 12

Band structure calculations for different twist angles without (left subpanels) and with (right subpanels) hBN alignment.

Figure 13

Selection of commensurate moiré structures listed in Table 1. Shown is the tBLG moiré lattice (black) as well as the AA site of the graphene/hBN moiré lattice (blue dots). Depending on the choice of $l$ and $m$ we receive a commensurate structure for different orientations $\phi$ of the two moiré lattices with respect to each other. The unit cell of the resulting supermoiré lattice is depicted as a red rhombus. The black scale bar corresponds to the spacing of the tBLG AA sites.