Metal-Insulator Transition in Graphene on Boron Nitride

Electrons in graphene aligned with hexagonal boron nitride are modeled by Dirac fermions in a correlated random-mass landscape subject to a scalar- and vector-potential disorder. We find that the system is insulating in the commensurate phase since the average mass deviates from zero. At the transition the mean mass is vanishing and electronic conduction in a finite sample can be described by a critical percolation along zero-mass lines. In this case graphene at the Dirac point is in a critical state with the conductivity $\sqrt{3}{e}^2/h$. In the incommensurate phase the system behaves as a symplectic metal.

Figure 1

(a) Flow diagram for the model (1) with random $V$, $\mathit{A}$, and $\mathit{m}$ (class $A$) [29]. Conductivity flows to quantum Hall critical point ${\sigma }_{xx}^{\mathrm{QHE}}\approx 0.57\times 4e^2/h$ provided $⟨m⟩=0$ and to zero otherwise [24, 25, 30]. (b) Phase diagram in the class $D$ ($V$, $\mathit{A}=0$). If $\mathrm{\Delta }$ is smaller than a critical value, the conductivity flows to ${\sigma }_{xx}^D=4e^2/\pi h$ for $⟨m⟩=0$ and to zero otherwise [23, 25].

Figure 2

Percolation model. Hexagons are colored yellow with a probability $p$, and the boundary between white and yellow hexagons corresponds to a zero-mass line that supports two counterpropagating chiral edge states in different valleys. Each line connecting the leads contributes a conductance of $2e^2/h$. The statistics of the zero-mass lines is described by the Schramm-Loewner evolution.

Figure 3

The average number of zero-mode lines connecting the leads at $x=0$ and $x=L$ in the random walk model on a hexagonal lattice (see Fig. 2) with periodic boundary conditions in $y$. The dots represent numerical data obtained for $W/L=2$. Solid lines correspond to Eq. (4).