Orbital diamagnetism in multilayer graphenes: Systematic study with the effective mass approximation

We present a theoretical study on the orbital magnetism in multilayer graphenes within the effective mass approximation. The Hamiltonian and thus susceptibility can be decomposed into contributions from subsystems equivalent to monolayer or bilayer graphene. The monolayer-type subband exists only in odd layers and exhibits a delta-function susceptibility at ${\epsilon }_F=0$. The bilayer-type subband appearing in every layer number gives a singular structure in the vicinity of ${\epsilon }_F=0$ due to the trigonal warping as well as a logarithmic tail away from ${\epsilon }_F=0$. The integral of the susceptibility over energy is approximately given only by the layer number.

Figure 1

Atomic structure of multilayer graphene with $AB$ (Bernal) stacking.

Figure 2

(Top) Projected band structure of the sub-Hamiltonian [Eq. (23)] with $\lambda =2$ and ${\gamma }_3∕{\gamma }_0=0.1$. $\mid \epsilon \mid ={\epsilon }_{\mathrm{trig}}=0.02{\gamma }_1$ is shown as horizontal dotted lines. Right panel shows zoom out of the left. (Bottom) 3D plot of the lower second band around the band touching point. Four Fermi points at $\epsilon =0$ indicated by dots.

Figure 3

Band structures of multilayer graphenes, $N=2,3,4,5$ with ${\gamma }_3∕{\gamma }_0=0.1$, around the $K$ point (taken as origin) along the $k_x$ axis. Right panel shows a zoom out of the left. Numbers assigned to curves indicate $m$.

Figure 4

Susceptibility of the sub-Hamiltonian [Eq. (23)] in the multilayer graphene with ${\lambda }_{N,m}=2$, ${\gamma }_3∕{\gamma }_0=0.1$, and several disorder strengths $\Gamma$. Energy is scaled in units of ${\epsilon }_{\mathrm{trig}}=0.02{\gamma }_1$. Dashed curves show plots for ${\gamma }_3=0$. Inset at the top is a zoom out of the top panel $(\Gamma =0.02{\epsilon }_{\mathrm{trig}})$ with units of energy ${\gamma }_1$.

Figure 5

Susceptibility of multilayer graphenes with layer numbers $N=1–5$ plotted against the Fermi energy. Results shown for several disorder strengths specified by constant scattering rate $\Gamma$.