New Dirac points and multiple Landau level crossings in biased trilayer graphene

Recently a new high-mobility Dirac material, trilayer graphene, was realized experimentally. The band structure of $ABA$-stacked trilayer graphene consists of a monolayer-like and a bilayer-like pair of bands. Here we study electronic properties of $ABA$-stacked trilayer graphene biased by a perpendicular electric field. We find that the combination of the bias and trigonal warping gives rise to a set of new Dirac points: In each valley, seven species of Dirac fermions with small masses of order of a few meV emerge. The positions and masses of the emergent Dirac fermions are tunable by bias, and one group of Dirac fermions becomes massless at a certain bias value. Therefore, in contrast to bilayer graphene, the conductivity at the neutrality point is expected to show nonmonotonic behavior, becoming of the order of a few $e^2/h$ when some Dirac masses vanish. Further, we analyze the evolution of the Landau level spectrum as a function of bias. The emergence of new Dirac points in the band structure translates into new threefold-degenerate groups of Landau levels. This leads to an anomalous quantum Hall effect, in which some quantum Hall steps have a height of $3e^2/h$. At an intermediate bias, the degeneracies of all Landau levels get lifted, and in this regime all quantum Hall plateaus are spaced by $e^2/h$. Finally, we show that the pattern of Landau level crossings is very sensitive to certain band structure parameters, and can therefore provide a useful tool for determining their precise values.

Figure 1

Schematic view of the lattice structure of the $ABA$-stacked trilayer graphene (left) and values of corresponding tight-binding parameters adopted from Ref. 5 (right). Red (blue) atoms belong to $A$ ($B$) sublattice of the corresponding layer. Next to each layer, the potential energy of electrons expressed via parameters ${\Delta }_1$ and ${\Delta }_2$ is shown.

Figure 2

Evolution of the band structure of trilayer graphene with bias ${\Delta }_1$. The bands in the vicinity of $K_{+}$ point in the Brillouin zone are shown as a function of $ka$, where $\mathit{k}$ describes momentum relative to the $K_{+}$ point. Solid (dashed) lines correspond to $\mathit{k}$ parallel to the $x$ axis ($y$ axis). At zero bias [panel (a)], red and blue lines describe monolayer-like and bilayer-like bands. At nonzero bias the monolayer-like and bilayer-like bands hybridize and no clear distinction can be made between them. As a result of this hybridization, one pair of bands moves to higher energies. The other pair of bands undergoes a series of transformations which lead to seven emergent Dirac points at higher bias ${\Delta }_1\gtrsim 0.1\mathrm{meV}$. The positions of these Dirac points in the Brillouin zone are illustrated in panel (g).

Figure 3

Evolution of the mass of emergent Dirac fermions as a function of bias ${\Delta }_1=0.05...0.25\mathrm{eV}$ for central DP (purple), first group of off-center DPs (dotted green), and second group of off-center DPs (dashed green). The mass of the second group of DPs vanishes at ${\Delta }_1\approx 0.185\mathrm{eV}$ and becomes negative at a larger bias.

Figure 4

Landau level spectrum as a function of magnetic field and bias. Plot (a) shows LL spectrum as a function of magnetic field $B=2...10$ T at zero displacement field, ${\Delta }_1=0$, and ${\Delta }_2=0$. Landau levels from the monolayer-like and bilayer-like block are shown in red and blue color; solid (dashed) lines correspond to $K_{+}$ ($K_{-}$) valleys. LLs are labeled with their indices, e.g., blue $1_{+}$, red $1_{+}$ ($1_{+}^{\mathrm{b}},1_{+}^{\mathrm{m}}$ in the main text) stand for LL with index 1 from $K_{+}$ valley of bilayer and monolayer sector, respectively. Lifting of spin degeneracy by the Zeeman field is not shown, so that every LL is doubly degenerate. Neutrality point is located between blue $0_{+}$ and blue $1_{+}$ LLs. Plots (b) and (c) show Landau levels at $B=5$ T and $B=10$ T as a function of displacement field. Red (blue) color is used to indicate LLs which belong to monolayer (bilayer) sector at ${\Delta }_1=0$.

Figure 5

Influence of the value of ${\gamma }_3$ on the pattern of avoided crossings between monolayer-like and bilayer-like LL. Magnetic field is $B=10$ T. For clarity, only LLs in the $K_{+}$ valley are shown. Plot (a) is for ${\gamma }_3=0.39$ eV used in this paper, whereas plot (b) is for twice larger value of ${\gamma }_3$. Labeling scheme of LLs is explained in the caption of Fig. 4.

Figure 6

Landau levels as a function of displacement field for different values of ${\Delta }_2$. Magnetic field $B=10$ T. Plot (a) uses zero value of ${\Delta }_2$; for plots (b) and (c) ${\Delta }_2=3$ meV and ${\Delta }_2=7$ meV, respectively. Value of ${\Delta }_2=7$ meV is sufficient to switch the order of blue ${(1,0)}_{-}$ and blue ${(1,0)}_{+}$ LLs from bilayer sector.