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Quantum Hall effect and the topological number in graphene

2006; American Physical Society; Volume: 74; Issue: 15 Linguagem: Inglês

10.1103/physrevb.74.155415

1550-235X

Yasumasa Hasegawa, Mahito Kohmoto,

Quantum and electron transport phenomena

Recently, an unusual integer quantum Hall effect was observed in graphene in which the Hall conductivity is quantized as ${\ensuremath{\sigma}}_{xy}=(\ifmmode\pm\else\textpm\fi{}2,\ifmmode\pm\else\textpm\fi{}6,\ifmmode\pm\else\textpm\fi{}10,\dots{})\ifmmode\times\else\texttimes\fi{}{e}^{2}∕h$, where $e$ is the electron charge and $h$ is the Planck constant. To explain this we consider the energy structure as a function of magnetic field (the Hofstadter butterfly diagram) on the honeycomb lattice and the Streda formula for Hall conductivity. The quantized Hall conductivities are identified as the topological TKNN integers [D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982); M. Kohmoto, Ann. Phys. (N.Y.) 160, 343 (1985)]. They are odd integers $\ifmmode\pm\else\textpm\fi{}1,\ifmmode\pm\else\textpm\fi{}3,\ifmmode\pm\else\textpm\fi{}5,\dots{}\ifmmode\times\else\texttimes\fi{}2$ (spin degrees of freedom) when a uniform magnetic field is as high as $30\phantom{\rule{0.3em}{0ex}}\mathrm{T}$ for example. The gaps corresponding to even integers, $\ifmmode\pm\else\textpm\fi{}2,\ifmmode\pm\else\textpm\fi{}4,\ifmmode\pm\else\textpm\fi{}6,\dots{}$ are too small to be observed, but when the system is anisotropic, which is described by the generalized honeycomb lattice, and/or in an extremely strong magnetic field, quantization in even integers takes place as well. We also compare the results with those for the square lattice in an extremely strong magnetic field.