Obstruction and Interference in Low-Energy Models for Twisted Bilayer Graphene

The electronic bands of twisted bilayer graphene (TBLG) with a large-period moiré superlattice fracture to form narrow Bloch minibands that are spectrally isolated by forbidden energy gaps from remote dispersive bands. When these gaps are sufficiently large, one can study a band-projected Hamiltonian that correctly represents the dynamics within the minibands. This inevitably introduces nontrivial geometrical constraints that arise from the assumed form of the projection. Here we show that this choice has a profound consequence in a low-energy experimentally observable signature that therefore can be used to tightly constrain the analytic form of the appropriate low-energy theory. We find that this can be accomplished by a careful analysis of the electron density produced by backscattering of Bloch waves from an impurity potential localized on the moiré superlattice scale. We provide numerical estimates of the effect that can guide experimental work to clearly discriminate between competing models for the low-energy band structure.

Figure 1

(a) Honeycomb moiré lattice structure of TBLG, spanned by ${\mathbf{L}}_1^M$ and ${\mathbf{L}}_2^M$, is formed by starting with an $AA$-stacked bilayer system and then twisting one layer relative to the other by an angle $\theta$. The resulting structure is a long-wavelength moiré pattern that has $AA$ regions (yellow-shaded), $AB$ regions (red-shaded), and $BA$ regions (blue-shaded). (b) Mini-Brillouin zone (MBZ) of TBLG formed by the momentum-mismatch of the original Brillouin zones. The blue and red hexagons show the monolayer BZs rotating in opposite directions. The MBZ is shown in green with high-symmetry points $\overline{K}$, ${\overline{K}}^{\prime }$, $\overline{M}$, and $\overline{\mathrm{\Gamma }}$, and reciprocal lattice vectors ${\mathbf{G}}_1^M$ and ${\mathbf{G}}_2^M$. (c)–(d) Band structure of TBLG numerically calculated from the continuum model with $\theta =2°$, $w_{AA}=79.7\mathrm{meV}$, $w_{AB}=97.5\mathrm{meV}$, and $\hslash v_F/a=2135.4\mathrm{meV}$. We also apply an interlayer bias $V=200\mathrm{meV}$ to shift the Dirac cones to improve visibility. The dash and solid lines are energy bands for the $-$ and $+$ valleys, respectively. We observe two Dirac cones in each valley at $\overline{K}$ and ${\overline{K}}^{\prime }$. We indicate the chirality of the band crossings schematically by blue and red cones. Dirac cones in the same valley have the same chirality in (c), corresponding to the topology of the continuum model, and opposite chirality in (d), corresponding to the topology of a two-orbital tight-binding model.

Figure 2

(a) Schematic representation of a possible experimental setup in which an impurity is placed atop an $AB$ region. (b),(c) The spatial distribution of the two Wannier orbitals in a single valley of TBLG. $|W_1⟩$ and $|W_2⟩$ are centered at an $AB$ and $BA$ region, respectively; however, most of their density is concentrated at the adjacent $AA$ regions. (d),(e) The LDOS simulated from inter-Dirac-cone scatterings at a bias energy $E=10\mathrm{meV}$ and $E=20\mathrm{meV}$ for the Wannier-representable model where the chirality at the two Dirac cones is opposite. We observe two wave front dislocations in the interference pattern. In addition, we also observe that as the probing energy increases, the period of radial oscillations decreases. (f),(g) The LDOS simulated from inter-Dirac-cone scatterings for the Wannier-obstructed model where the chirality at the two Dirac cones is opposite. Unlike before, here, we observe no wave front dislocations.

Figure 3

Each row shows the change in LDOS induced by an impurity placed on an $AB$ region (top row) or an $AA$ region (bottom row). (b) and (h) show the full interference pattern, and (c) and (i) show the magnitudes of their fast Fourier transforms (FFT). From the FFT, we filter out time-reversed pairs of momenta in three independent directions, and then calculate the inverse FFT to observe the density wave fronts. Panels (d)–(f) and (j)–(i) show the wave front interference pattern after applying corresponding FFT filters indicated on the insets. As can clearly be seen, (d)–(f) feature one wave front dislocation. On the contrary, (j)–(l) do not have any dislocation.