Electronic transport in disordered bilayer and trilayer graphene

We present a detailed numerical study of the electronic transport properties of bilayer and trilayer graphene within a framework of single-electron tight-binding model. Various types of disorder are considered, such as resonant (hydrogen) impurities, vacancies, short- or long-range Gaussian random potentials, and Gaussian random nearest-neighbor hopping. The algorithms are based on the numerical solution of the time-dependent Schrödinger equation and applied to calculate the density of states and conductivities (via the Kubo formula) of large samples containing millions of atoms. In the cases under consideration, far enough from the neutrality point, depending on the strength of disorders and the stacking sequence, a linear or sublinear electron-density-dependent conductivity is found. The minimum conductivity ${\sigma }_{\text{min}}\approx 2e^2/h$ (per layer) at the charge neutrality point is the same for bilayer and trilayer graphene, independent of the type of the impurities, but the plateau of minimum conductivity around the neutrality point is only observed in the presence of resonant impurities or vacancies, originating from the formation of the impurity band.

Figure 1

Atomic structure of bilayer, $ABA$- and $ABC$-stacked trilayer graphene.

Figure 2

(Color online) Top panel: DOS of bilayer graphene $\left({\gamma }_1={\gamma }_3=0.1t\right)$ with different concentrations of resonant impurities $\left({\epsilon }_d=-t/16,V=2t\right)$ added on both layers. Middle panel: comparison of the conductivity of the BLG (line) and SLG (square) with the same concentration of resonant impurities. Bottom panel: comparison of the conductivity of bilayer graphene $\left({\gamma }_1={\gamma }_3=0.1t\right)$ with the same amount of resonant impurities $\left({\epsilon }_d=-t/16,V=2t\right)$ added on both layers (line, $n_i$) or only one layer (triangle, $n_{1i}$). Each layer in BLG contains $4096\times 4096$ carbon atoms, and SLG contains $6400\times 6400$ carbon atoms.

Figure 3

(Color online) Comparison of the DOS and conductivity of the SLG, BLG, TLG, and QLG with the same concentration of resonant impurities $\left(n_i=0.5\mathrm{\%},{\epsilon }_d=-t/16,V=2t\right)$. ${\gamma }_1={\gamma }_3=0.1t$ in top panels and ${\gamma }_1=0.5t,{\gamma }_3=0.1t$ in bottom panels. SLG contains $6400\times 6400$ carbon atoms, each layer in BLG, TLG, and QLG contains $4096\times 4096$, $3200\times 3200$, and $2400\times 2400$ carbon atoms, respectively.

Figure 4

(Color online) Comparison of the DOS of bilayer graphene with different interlayer interactions: ${\gamma }_1=0.1t$, $0.2t$, and $0.5t$ (${\gamma }_3$ is fixed as $0.1t$). Inner panel: normalized DOS and energy in units of $1/{\gamma }_1$ and ${\gamma }_1$. Each layer in BLG contains $4096\times 4096$ carbon atoms.

Figure 5

(Color online) DOS and conductivity of bilayer graphene with resonant impurities $\left({\epsilon }_d=-t/16,V=2t\right)$ added on both layers. The interlayer parameter ${\gamma }_1$ is fixed as $0.5t$, and ${\gamma }_3=0$ (black line), $0.1t$ (red dashed line), and $0.5t$ (green dot line). Each layer contains $4096\times 4096$ carbon atoms.

Figure 6

(Color online) Comparison of the DOS and conductivity of the SLG, BLG, and TLG with the same concentration of vacancies $\left(n_x=0.5\mathrm{\%}\right)$. The parameters of the interlayer coupling are ${\gamma }_1=0.5t$ and ${\gamma }_3=0.1t$. SLG contains $6400\times 6400$ carbon atoms, each layer in BLG and TLG contains $4096\times 4096$ and $3200\times 3200$ carbon atoms (sites), respectively.

Figure 7

(Color online) Contour plot of the on-site potentials in the central part of a graphene layer $\left(4096\times 4096\right)$ with short-range $\left(\Delta =3t,d=0.65a,P_v=0.5\mathrm{\%}\right)$ or long-range $\left(\Delta =1t,d=5a,P_v=0.1\mathrm{\%}\right)$ Gaussian potential.

Figure 8

(Color online) DOS and conductivity of bilayer graphene $\left({\gamma }_1={\gamma }_3=0.1t\right)$ with short-range $\left(\Delta =3t,d=0.65a\right)$ or long-range $\left(\Delta =1t,d=5a\right)$ Gaussian potential. Each layer contains $4096\times 4096$ carbon atoms.

Figure 9

(Color online) DOS and conductivity of bilayer graphene $\left({\gamma }_1=0.5t,{\gamma }_3=0.1t\right)$ with long-range $\left(\Delta =1t,d=5a\right)$ or short-range $\left(\Delta =3t,d=0.65a\right)$ Gaussian potential. Each layer contains $4096\times 4096$ carbon atoms.

Figure 10

(Color online) DOS and conductivity of bilayer graphene $\left({\gamma }_1={\gamma }_3=0.1t\right)$ with short-range $\left({\Delta }_t=t,d_t=0.65a\right)$ or long-range $\left({\Delta }_t=0.5t,d_t=5a\right)$ Gaussian hopping. Each layer contains $4096\times 4096$ carbon atoms.

Figure 11

(Color online) DOS and conductivity of bilayer graphene $\left({\gamma }_1={\gamma }_3=0.1t\right)$ with short-range $\left({\Delta }_t=t,d_t=0.65a\right)$ or long-range $\left({\Delta }_t=0.5t,d_t=5a\right)$ Gaussian hopping. Each layer contains $4096\times 4096$ carbon atoms.