Graphene is one of the solid phases of carbon, appearing as a single-layer honeycomb lattice. This may be planar or tubular, etc.

Graphene is a prime example (and among the first to be discovered) of a topological phase of matter. Specifically:

  1. at coarse resolution (currently accessible to experiment) it appears as a (time-reversal and space inversion-symmetric) topological semi-metal, due to a gap between its valence band and conduction band which closes only over two Dirac points in its Brillouin torus;

  2. at finer resolution – namely when the spin-orbit coupling of the electrons is resolved (thought to be 10 3\sim10^{-3}\,meV [[MHSSKM06]] and hence too small for experimental observation, currently), which reveals a small energy gap opening at the two would-be Dirac points – graphene appears as a (time-reversal symmetric) topological insulator (Kane & Mele 05a), whose non-trivial topological phase is witnessed by the non-trivial Kane-Mele invariant in / 2 \mathbb{Z}/2 (Kane & Mele 05b).


The electronic band structure of graphene (reviewed in WZLLJHD 12) was predicted (long before the term was coined) already in

The synthesis/detection of graphene is due to

(The procedure, won a Nobel Prize and the authors made it freely available without patenting.)

Computation of the (tiny) spin-orbit coupling in graphene:

  • Hongki Min, J. E. Hill, N. A. Sinitsyn, B. R. Sahu, Leonard Kleinman, A. H. MacDonald, Intrinsic and Rashba spin-orbit interactions in graphene sheets, Phys. Rev. B 74 (2006) 165310 [[doi:10.1103/PhysRevB.74.165310]]

Observation that the spin-orbit coupling in graphene should open the gap at the Dirac points revealing a quantum spin Hall effect in graphene:


  • Nathan Weiss, Hailong Zhou, Lei Liao, Yuan Liu, Shan Jiang, Yu Huang, Xiangfeng Duan, Graphene: an emerging electronic material, Adv Mater. 24 (43) (2012) 5782-825 (doi:10.1002/adma.201201482)

  • Wikipedia, Graphene

Review in the context of topological phases of matter and specifically topological semi-metals:

The 2+1 dim Dirac equation is used in modeling graphene:

  • Konstantin Novoselov, Andre Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, A. A. Firsov, Two-dimensional gas of massless Dirac fermions in graphene, Nature 438, 197-200 (2005) doi

  • María A. H. Vozmediano, Renormalization group aspects of graphene, pdf

category: physics

Last revised on May 25, 2022 at 09:48:25. See the history of this page for a list of all contributions to it.