Gate-tunable topological flat bands in twisted monolayer-bilayer graphene

We investigate the band structure of twisted monolayer-bilayer graphene (tMBG) trilayers, or twisted graphene on bilayer graphene, as a function of twist angles and perpendicular electric fields in search of optimal conditions for achieving isolated nearly flat bands. Narrow bandwidths comparable to or smaller than the effective Coulomb energies satisfying $U_{\text{eff}}/W\gtrsim 1$ are expected for twist angles in the range of $0.3^{\circ }–1.5^{\circ }$, more specifically in islands around $\theta \sim 0.5^{\circ },0.{85}^{\circ },1.3^{\circ }$ for appropriate perpendicular electric field magnitudes and directions. The valley Chern numbers of the electron-hole asymmetric bands depend intrinsically on the details of the hopping terms in the bilayer graphene, and extrinsically on factors like electric fields or average staggered potentials in the graphene layer aligned with the contacting hexagonal boron nitride substrate. This tunability of the band isolation, bandwidth, and valley Chern numbers makes tMBG trilayers a more versatile system than twisted bilayer graphene for finding nearly flat bands prone to strong correlations.

Figure 1

Schematic diagram of twisted monolayer graphene stacked on top of a Bernal stacked bilayer graphene (tMBG). (a) The side view and (b) top view of the tMBG atomic structure. The $A_l$ and $B_l$ are two sublattices of $l\text{th}$ layer, $l$ = b (bottom), m (middle), and t (top). A relative twist angle $\theta$ between the monolayer and bilayer graphene gives rise to moiré pattern lattice vectors ${\tilde{L}}_i$. The mini-moiré Brillouin zone (MBZ) is illustrated by a black hexagon, along with the MBZ high symmetry points. The red and blue hexagons correspond to the first Brillouin zone of monolayer and bilayer graphene. (c) We schematically illustrate the effect of an applied electric field by means of the interlayer potentials $\pm \mathrm{\Delta }$ introduced between the bottom and top graphene layers, and the staggered potential ${\mathrm{\Delta }}_S$ in the top monolayer graphene due to alignment with hexagonal boron nitride giving rise to ${\mathrm{\Delta }}_S<0$ for BN alignment (for graphene's $A_t$ and $B_t$ sublattices aligned with boron and nitrogen sites) and to ${\mathrm{\Delta }}_S>0$ for NB alignments.

Figure 2

Variation of bandwidth as a function of twist angle for the valence and conduction bands within the minimal (top panel) and complete (bottom panel) models in twisted monolayer graphene on bilayer graphene (tMBG). We compare the bandwidth variations for applied potential $\mathrm{\Delta }$ between the layers. Extremely small bandwidths are associated to the twist angle $\theta =1.{06}^{\circ }$ in the minimal model for $\mathrm{\Delta }=0\mathrm{eV}$, which are a factor two smaller than the twisted bilayer graphene (tBG) with similar system parameters, where the dotted horizontal lines represent the bandwidth of tBG at the belly maximum. However, in the complete model the bandwidths increase due to the remote hopping terms.

Figure 3

Color map distribution of flat band bandwidth ($W$), the primary (${\delta }_p$) and secondary (${\delta }_s$) band gaps for the low-energy valence ($V$) and conduction ($C$) bands of tMBG in the parameter space of $\theta$ and $\mathrm{\Delta }$. The (a) and (b) panels show the phase diagrams obtained using the minimal and complete models, respectively, for the staggered potential ${\mathrm{\Delta }}_S=0\mathrm{eV}$ and the (c) and (d) panels show the results obtained using the complete model with the staggered potential ${\mathrm{\Delta }}_S=\pm 0.01\mathrm{eV}$.

Figure 4

Band structures for different values of interlayer potentials (a) $\mathrm{\Delta }=0\mathrm{eV}$, (b) $\mathrm{\Delta }=0.060\mathrm{eV}$, and (c) $\mathrm{\Delta }=-0.040\mathrm{eV}$ for a twist angle $\theta =1.{21}^{\circ }$ obtained using minimal/complete models illustrated by the dotted/solid lines together with the complete model's density of states $D\left(E\right)$, local density of states $D(\mathbf{r},E)$ for different local commensurate stacking regions ABB, ABC, and ABA, and their differences $\delta \tilde{D}(\mathbf{r},E)={\tilde{D}}_{\mathrm{\Delta }}(\mathbf{r},E)-{\tilde{D}}_0(\mathbf{r},E)$ for finite $\mathrm{\Delta }$ interlayer potentials, where $\tilde{D}(\mathbf{r},E)=D(\mathbf{r},E)/\max \left[\right|D(\mathbf{r},E)\left|\right]$ are the normalized LDOS evaluated at the van Hove singularity energies $E=E_{\mathrm{VHS}}$ of the nearly flat bands. The isolated low-energy bands of valence/conduction bands are represented by red/blue solid lines accompanied by their valley Chern numbers. On the right we represent the real space plot of normalized LDOS at the van Hove singularities (VHS) of conduction(blue)/valence(red) for $\mathrm{\Delta }=0\mathrm{eV}$, and their differences for $\mathrm{\Delta }=0.060\mathrm{eV}$ and $\mathrm{\Delta }=-0.040\mathrm{eV}$ that show sign dependent opposing contrasts for the conduction and valence bands.

Figure 5

Band structure and density of states (DOS) of tMBG for select twist angles $0.{51}^{\circ }, 0.{85}^{\circ }, 1.{21}^{\circ }, 1.{31}^{\circ }$, and $1.{41}^{\circ }$ and finite interlayer potential $\mathrm{\Delta }$ that maximizes correlations for either the conduction or valence bands such that $U_{\mathrm{eff}}/W>1$. We compare the band structures and DOS for given twist angles obtained using minimal (light blue) and complete (black) models. In each band structure within the complete model, the low energy valence band is identified with red and the conduction band with blue. The calculated Chern numbers within the complete model are shown together with the respective low energy bands using the same color.

Figure 6

Band structures and Berry curvatures of tMBG for the twist angle $\theta =1.{21}^{\circ }$ with applied interlayer potential $\mathrm{\Delta }=\pm 0.02\mathrm{eV}$. We compare the minimal (light blue) and complete (black) models for interlayer potential $\mathrm{\Delta }=-0.04,0.06\mathrm{eV}$ with (a) the staggered potential set to ${\mathrm{\Delta }}_S=0$, (b) ${\mathrm{\Delta }}_S=0.01\mathrm{eV}$, and (c) ${\mathrm{\Delta }}_S=-0.01\mathrm{eV}$ that do not change the valley Chern numbers for the specific cases explored. For each case of $\mathrm{\Delta }$ and ${\mathrm{\Delta }}_S$, we show the Berry curvatures plot, and in each surface plot, the first moiré BZ is indicated by black hexagonal lines. The Berry curvature hot spots are observed at ${\tilde{K}}^{\prime }$ and $\tilde{\mathrm{\Gamma }}$ within the MBZ.

Figure 7

Band structures and Berry curvatures of tMBG for the twist angle $\theta =1.{41}^{\circ }$ with applied electric field induced potentials of $\mathrm{\Delta }=\pm 0.02\mathrm{eV}$. We compare the minimal (light blue) and complete (black) models for electric field induced potentials $\mathrm{\Delta }=\pm 0.02\mathrm{eV}$ and (a) staggered potential set to ${\mathrm{\Delta }}_S=0$, for (b) nonzero staggered potential ${\mathrm{\Delta }}_S=0.01\mathrm{eV}$ and (c) ${\mathrm{\Delta }}_S=-0.01\mathrm{eV}$. The low energy bands show variations in the valley Chern numbers compared to the case when ${\mathrm{\Delta }}_S=0$. We show the Berry curvatures plot for each combination of $\mathrm{\Delta }$ and ${\mathrm{\Delta }}_S$. In each surface plot, the first moiré BZ is indicated by black hexagonal lines. The Berry curvature hot spots are observed at ${\tilde{K}}^{\prime }$ and this is due to the small gap opening near the Dirac cones of monolayer graphene in tMBG.

Figure 8

Band structures for opposite staggered potential signs ${\mathrm{\Delta }}_S=\pm 0.01\mathrm{eV}$ are shown on the left (a), (c), (d) and right (b), (e), (f) panels for a fixed twist angle $\theta =1.{07}^{\circ }$, close to the magic angle $\theta =1.{06}^{\circ }$ of the minimal model, for various interlayer potentials $\mathrm{\Delta }$. Similar plots can be found for (g), (i), (j) and (h), (k), (l) panels for twist angle $\theta =1.{21}^{\circ }$. Minimal and complete model bands are shown through dotted and solid lines, respectively, and for the complete model the density of states $D\left(E\right)$, the local density of states $D(\mathbf{r},E)$, and the difference $\delta \tilde{D}(\mathbf{r},E)={\tilde{D}}_{{\mathrm{\Delta }}_S}(\mathbf{r},E)-{\tilde{D}}_0(\mathbf{r},E)$ due to the staggering potential ${\mathrm{\Delta }}_S$, where $\tilde{D}(\mathbf{r},E)=D(\mathbf{r},E)/\max \left[\right|D(\mathbf{r},E)\left|\right]$ is the normalized LDOS evaluated at the van Hove singularity energies $E=E_{\mathrm{VHS}}$ of the nearly flat bands. The electronic wave functions are found to concentrate maximally for conduction bands at ABB stacking regions.

Figure 9

Sublattice and layer projected LDOS (in arb. units) at valence/conduction VHS for twist angle $\theta =1.{07}^{\circ }$, close to the magic angle $\theta =1.{06}^{\circ }$ of the minimal model, calculated for the complete Hamiltonian that includes the remote hopping terms and using different combinations of $\mathrm{\Delta }=0.0,0.047,-0.032\mathrm{eV}$ and ${\mathrm{\Delta }}_S=0.0,0.01,-0.01\mathrm{eV}$. The low energy sublattices (${\mathrm{B}}_m$ of the bilayer and ${\mathrm{B}}_t$ of monolayer) are found to concentrate most of the flat bands charge density at the van Hove singularity.

Figure 10

Phase diagram of ratio between the effective screened Coulomb potential $U_{\mathrm{eff}}$ and bandwidth $W$ ($U_{\mathrm{eff}}/W$) as a function of $\theta$ and $\mathrm{\Delta }$ in tMBG, where $U_{\mathrm{eff}}/W>1$ indicates the Coulomb-interaction driven ordered phases. Panels (a) and (b) compare the $U_{\mathrm{eff}}/W$ ratio obtained using minimal and complete model. The minimal model is based on the same complete Hamiltonian but we set to zero the remote hopping terms $t_3, t_4$ as well as the site potential differences $\delta$ in the bilayer Hamiltonian part. The unequal color maps of $U_{\mathrm{eff}}/W$ between the valence and conduction bands indicate the strong particle-hole asymmetry in tMBG. The complete model reduces the area of $U_{\mathrm{eff}}/W>1$ when compared to the minimal model. The color map distribution of $U_{\mathrm{eff}}/W$ is changed slightly with the staggered potential ${\mathrm{\Delta }}_S>0$, (c) ${\mathrm{\Delta }}_S=0.01\mathrm{eV}$, and (d) ${\mathrm{\Delta }}_S=-0.01\mathrm{eV}$. The area of $U_{\mathrm{eff}}/W>1$ is enlarged near twist angle of $\theta \approx 0.5^{\circ }$ and $\theta \approx 1.2^{\circ }$ for ${\mathrm{\Delta }}_S>0$ when compared to the ${\mathrm{\Delta }}_S=0$ case within the complete remote hopping parameters model; meanwhile, the area $U_{\mathrm{eff}}/W>1$ is reduced when ${\mathrm{\Delta }}_S<0$.

Figure 11

Phase diagram of the valley Chern numbers in tMBG using the complete model for the low-energy conduction and valence bands as a function of twist angle $\theta$ and interlayer potentials $\mathrm{\Delta }$ for three values of sublattice staggering potential ${\mathrm{\Delta }}_S$ in the monolayer. The different valley Chern numbers ($C=0,\pm 1,\pm 2,\pm 3$) for the valence/conduction bands as a function electric field induced interlayer potential $\mathrm{\Delta }$ indicate the possibility of achieving topological phase transitions for various system parameters. The large $C=\pm 3$ phases disappear for ${\mathrm{\Delta }}_S\ne 0$. The black dashed lines enclose the respective strong correlation regions satisfying $U_{\mathrm{eff}}/W>1$ in Fig. 10.