Validity of the two-component model of bilayer and trilayer graphene in a magnetic field

The eigenstates of an electron in the chiral two-dimensional electron gas (C2DEG) formed in an AB-stacked bilayer or an ABC-stacked trilayer graphene is a spinor with four or six components, respectively. These components give the amplitude of the wave function on the four or six carbon sites in the unit cell of the lattice. In the tight-binding approximation, the eigenenergies are thus found by diagonalizing a $4\times 4$ or a $6\times 6$ matrix. In the continuum approximation where the electron wave vector $k\ll 1/a_0,$ with $a_0$ the lattice constant of the graphene sheets, a common approximation is the two-component (or “two-band”) model1 where the eigenstates for the bilayer and trilayer systems are described by a two-component spinor that gives the amplitude of the wave function on the two sites with low energy $\left|E\right|\ll {\gamma }_1$ where ${\gamma }_1$ is the hopping energy between sites that are directly above one another in adjacent layers. The two-component model has been used extensively to study the phase diagram of the C2DEG in a magnetic field as well as its transport and optical properties. In this paper, we use a numerical approach to compute the eigenstates and Landau level energies of the full tight-binding model in the continuum approximation and compare them with the prediction of the two-component model when the magnetic field or an electrical bias between the outermost layers is varied. Our numerical analysis shows that the two-component model is a good approximation for bilayer graphene in a wide range of magnetic field and bias but mostly for Landau level $M=0.$ The applicability of the two-component model in trilayer graphene, even for level $M=0,$ is much more restricted. In this case, the two-component model fails to reproduce some of the level crossings that occur between the sublevels of $M=0.$

Figure 1

Schematic drawing of the band structure of AB-stacked bilayer graphene (left) and ABC-stacked trilayer graphene (right) in the minimal model for wave vector $\mathbf{k}$ near the valley point $K.$

Figure 2

Crystal structure of the (a) Bernal-stacked bilayer graphene and (b) ABC-stacked trilayer graphene. Definition of the hopping parameters for the: (c) bilayer and (d) trilayer systems.

Figure 3

Landau level spectrum as a function of the ratio ${\gamma }_3/{\gamma }_0$ at zero bias in the four-band (lines) and two-band (blue squares) model of BLG for (a) $B=0.1$ T and (b) $B=1.0$ T. The two-band model considered in this figure includes the warping term ${\gamma }_3.$

Figure 4

Landau level spectrum in TLG at zero bias in the six-band model as a function of the ratio $|{\gamma }_3/{\gamma }_0|$ with ${\gamma }_2=0$ for (a) $B=1$ T and (b) $B=10$ T and as a function of the ratio $|{\gamma }_2/{\gamma }_0|$ with ${\gamma }_3=0$ for (c) $B=1$ T and (d) $B=10$ T.

Figure 5

Behavior of the Landau levels with magnetic field at zero bias for valley $K_{+}$ for the four-band model (lines) and two-band model without ${\gamma }_2$ and ${\gamma }_3$ (blue squares). (a) bilayer graphene; (b) low-energy sector showing the $M=0$ levels of (a); (c) trilayer graphene; (d) low-energy sector showing the $M=0$ levels of (c).

Figure 6

Dispersion of the first few Landau levels with bias for the four-band model (lines) and two-band model without ${\gamma }_3$ (blue squares) at magnetic field $B=10$ T and $B=30$ T for bilayer graphene.

Figure 7

Dispersion of the first few Landau levels with bias for the six-band model (lines) and two-band model without ${\gamma }_2,{\gamma }_3$ (blue squares) at magnetic field $B=10$ T and $B=30$ T for trilayer graphene.

Figure 8

Dispersion of the three Landau levels $n=0,1,2$ in $M=0$ with bias in trilayer graphene at $B=30$ T showing the multiple crossings in the 6BM (black lines). The symbols are for the 2BM without ${\gamma }_2,{\gamma }_3.$

Figure 9

Dispersion of the Landau levels with magnetic field at zero bias in the six-band model of trilayer graphene: (a) minimal model, (b) with the addition of ${\delta }_0,$ (c) with the further addition of ${\delta }_0$ and ${\gamma }_4$, and (d) with the further addition of ${\delta }_0,{\gamma }_4$, and ${\gamma }_2.$

Figure 10

Low-energy eigenstates of the four-band model and the probabilities $P$ and $\Lambda$ as a function of bias for the $M=0;n=0,1$ levels of bilayer graphene at (a) $B=10$ T and (b) $B=30$ T.

Figure 11

Low-energy eigenstates of the six-band model and the probabilities $P$ and $\Lambda$ as a function of bias for the $M=0;n=0,1,2$ levels of trilayer graphene at (a) $B=10$ T and (b) $B=30$ T.