Electric transport and magnetic properties in multilayer graphene

We discuss electric transport and orbital magnetism of multilayer graphenes in a weak-magnetic field using the matrix decomposition technique. At zero temperature, the minimum conductivity is given by that of the monolayer system multiplied by the layer number $N$, independent of the interlayer hopping $t$. When the interlayer hopping satisfies the condition $t⪢\hslash ∕\tau$ with $\tau$ being collision time of impurity scattering, $[N∕2]$ kinks and $[N∕2]+1$ plateaux appear in the Fermi-energy (gate voltage) dependence of the conductivity and the Hall conductivity, respectively. These behaviors are interpreted as multiband effects. We also found that the Hall conductivity and the magnetic susceptibility take minimum value as a function of temperature, for certain value of the gate voltage. This behavior is explained by Fermi-energy dependence of these functions at zero temperature.

Figure 1

(Color online) Two stacking structures of multilayer graphene.

Figure 2

(Color online) Band structure of multilayer systems $(N=1–5)$ with the Bernal stacking [Eq. (1), solid line] and with the rhombohedral stacking [Eq. (2), dashed line]. For the Bernal stacking, linear dispersion relations appear for odd layer cases. The dispersion relation of the effective Hamiltonian of the rhombohedral stacking [Eq. (3), dotted line] is also shown.

Figure 3

(Color online) Fermi-energy $(\mu ∕\Gamma )$ dependence of the longitudinal conductivity of Bernal stacking systems with $N=2,3,4,5$, scaled by the minimum conductivity of the monolayer system ${\sigma }_{\min }=e^2∕\pi h$. In these results, $[N∕2]$ kinks appear for the positive (or negative) Fermi-energy regions.

Figure 4

(Color online) Comparison between the longitudinal conductivity obtained by the linear response theory (solid lines) and that by the semiclassical analysis (dashed lines) for one to five layer systems with $t∕\Gamma =6$.

Figure 5

(Color online) Fermi-energy $(\mu ∕\Gamma )$ dependence of the Hall conductivity of Bernal stacking systems with $N=2,3,4,5$, scaled by the classical Hall conductivity of the monolayer system ${\omega }_c\tau {\sigma }_0$. In these results, $[N∕2]+1$ plateaux appear in large $t∕\Gamma$ cases.

Figure 6

(Color online) Comparison between the Hall conductivity obtained by the linear response theory (solid lines) and that by the semiclassical analysis (dashed lines) for one to five layer systems with $t∕\Gamma =6$.

Figure 7

(Color online) Fermi-energy $(\mu ∕\Gamma )$ dependence of the Hall conductivity of a rhombohedral stacking system with $N=3$. In this case, the first plateau appears at ${\sigma }_{xy}\simeq \pm {\omega }_c\tau {\sigma }_0$, reflecting the band structure.

Figure 8

(Color online) Magnetic susceptibility of the bilayer system at zero temperature as a function of the Fermi energy $\mu$ for (a) fixed interlayer hopping $t=1$ and for (b) fixed scattering rate $\Gamma =1$. These data are scaled by ${\chi }_0=-6e^2{v}^2∕{\pi }^2{c}^2$. $t$ dependence of the susceptibility is similar to that of gap dependence in the monolayer system [see Fig. 1(a) of Ref. 27].

Figure 9

(Color online) Temperature dependence of the magnetic susceptibility $\chi$ of (a) the monolayer system and (b) the bilayer system for several strengths of the interlayer hopping at $\mu ∕\Gamma =1.5$. A minimum value appears as a function of temperature.

Figure 10

(Color online) Temperature dependence of the Hall conductivity ${\sigma }_{xy}$ of (a) the monolayer system and (b) the bilayer system for several strengths of the interlayer hopping at $\mu ∕\Gamma =3$. A minimum value appears as a function of temperature.

Figure 11

(Color online) Temperature dependence of the conductivity ${\sigma }_{xx}$ of (a) the monolayer system and (b) the bilayer system for several strengths of the interlayer hopping at $\mu ∕\Gamma =3$. They behave monotonically as a function of temperature.