Magnetization of graphene tubules

Magnetization (M) comes from both the persistent currents and the spin polarization. The spin-B interaction is important in a graphene tubule, because it makes the one-dimensional subband with the divergent density of states capable of crossing the Fermi level (${\mathit{E}}_{\mathit{F}}$=0 eV). It causes cusps in magnetization and power divergencies in differential susceptibility (${\mathrm{\chi }}_{\mathit{M}}$), and destroys the periodicity (period ${\mathrm{\phi }}_0$=hc/e) of the physical properties. The special structures shown in M and ${\mathrm{\chi }}_{\mathit{M}}$ are found to be insensitive to the chirality. The power divergencies in ${\mathrm{\chi }}_{\mathit{M}}$ are replaced by the peak structures at low temperature (T). The order of ${\mathrm{\chi }}_{\mathit{M}}$ is ${10}^{\mathrm{-}4}$–${10}^{\mathrm{-}5}$; therefore, the peak structures are measurable at T≤1 K. The temperature effect in reducing magnetization is relatively obvious for a larger semiconducting tubule. Moreover, the anomalous temperature effect due to the spin-B interaction exists in all the metallic tubules at the relatively low T. For the doped graphene tubule, M and ${\mathrm{\chi }}_{\mathit{M}}$ exhibit more special structures, since both the electronic structure and the finite Fermi level vary with φ simultaneously. The magnetic response is enhanced by the doping, and it is relatively strong for a larger tubule. The magnetism at the small flux is possibly altered from paramagnetism (diamagnetism) to diamagnetism (paramagnetism) by varying the free-carrier density.