nLab graphene

Idea

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 $\sim10^{-3}\,$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 $\mathbb{Z}/2$ (Kane & Mele 05b).

Related concepts

References

General

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

• Philip Russel Wallace, The Band Theory of Graphite, Phys. Rev. 71 (1947) 622 (doi:10.1103/PhysRev.71.622)

The synthesis/detection of graphene is due to

• Konstantin Novoselov, Andre Geim, S. V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A.A. Firsov, Electric field effect in atomically thin carbon films, Science 306, no. 5696, pp. 666-669 (2004) (doi:10.1126/science.1102896, arXiv:cond-mat/0410550)

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

Further discussion of the electron band structure of graphene:

• A. H. Castro Neto, F. Guinea, N. M. R. Peres, Konstantin S. Novoselov, and Andre K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81 (2009) 109 $[$doi:10.1103/RevModPhys.81.109$]$

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:

• Charles Kane, Eugene Mele, Quantum Spin Hall Effect in Graphene, Phys. Rev. Lett. 95, 226801 (2005) (arXiv:cond-mat/0411737, doi:10.1103/PhysRevLett.95.226801)

• Charles Kane, Eugene Mele, $Z_2$ Topological Order and the Quantum Spin Hall Effect, Phys. Rev. Lett. 95 (2005) 146802 (doi:10.1103/PhysRevLett.95.146802)

Review:

• 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:

• Jérôme Cayssol, Jean-Noël Fuchs, Section 6 of: Topological and geometrical aspects of band theory, J. Phys. Mater. 4 (2021) 034007 (arXiv:2012.11941, doi:10.1088/2515-7639/abf0b5)

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

On non-perturbative effects in graphene:

• Juan Angel Casimiro Olivares, Ana Julia Mizher, Alfredo Raya, Non-perturbative field theoretical aspects of graphene and related systems [arXi:2109.10420]

Discussion via AdS-CFT in condensed matter physics:

• Jeong-Won Seo, Taewon Yuk, Young-Kwon Han, Sang-Jin Sin, ABC-stacked multilayer graphene in holography [arXiv:2208.14642]

On fractional charges in graphene:

• Chang-Yu Hou, Claudio Chamon, Christopher Mudry, Electron fractionalization in two-dimensional graphenelike structures, Phys. Rev. Lett. 98 (2007) 186809 [arXiv:cond-mat/0609740, doi:10.1103/PhysRevLett.98.186809]

Berry connection

On “fictitious” contributions to the Berry connection on the Brillouin torus of graphene:

• Mircea Trif, Pramey Upadhyaya, Yaroslav Tserkovnyak, Theory of electromechanical coupling in dynamical graphene, Phys. Rev. B 88 245423 (2013) $[$doi:10.1103/PhysRevB.88.245423, arXiv:1210.7384$]$

Movement of Dirac points

Moving the nodal points in graphene(-variants) by changing external parameters such as lattice anisotropy or strain (see also further discussion of external manipulation by strain here, and see the references on momentum-space braiding of band nodes):

• Cui-Lian Lia, New position of Dirac points in the strained graphene reciprocal lattice, AIP Advances 4 (2014) 087119 [doi:10.1063/1.4893239]

• Marc Dvoraka, Zhigang Wu, Dirac point movement and topological phase transition in patterned graphene, Nanoscale 7 (2015) 3645-3650 [doi:10.1039/C4NR06454B]

• Zhenzhu Li, Zhongfan Liu, Zhirong Liu: Movement of Dirac points and band gaps in graphyne under rotating strain, Nano Research 10 (2017) 2005–2020 [doi:10.1007/s12274-016-1388-z]

• Jian Kang, Oskar Vafek, Non-Abelian Dirac node braiding and near-degeneracy of correlated phases at odd integer filling in magic angle twisted bilayer graphene, Phys. Rev. B 102 (2020) 035161 [arXiv:2002.10360, doi:10.1103/PhysRevB.102.035161]

Movement of Dirac points in a cousin of the Haldane model:

• Miguel Gonçalves, Pedro Ribeiro, Eduardo V. Castro, Dirac points merging and wandering in a model Chern insulator, Europhysics Letters 124 6 (2018) [doi:10.1209/0295-5075/124/67003]

and in photonic crystal-analogs:

• Yong-Heng Lu et al., Observing movement of Dirac cones from single-photon dynamics, Phys. Rev. B 103 064304 (2021) [doi:10.1103/PhysRevB.103.064304]

Holographic description

On AdS-CMT duality for graphene-like systems via full supergravity (retaining the gravitino):

• Laura Andrianopoli, Bianca L. Cerchiai, Riccardo D'Auria, Mario Trigiante, Unconventional Supersymmetry at the Boundary of $AdS_4$ Supergravity, J. High Energ. Phys. 2018 7 (2018) [arXiv:1801.08081, doi:10.1007/JHEP04(2018)007]

• Laura Andrianopoli, Bianca L. Cerchiai, Riccardo D'Auria, A. Gallerati, R. Noris, Mario Trigiante, Jorge Zanelli, $\mathcal{N}$-Extended $D=4$ Supergravity, Unconventional SUSY and Graphene, JHEP 01 (2020) 084 [arXiv:1910.03508, doi:10.1007/JHEP01(2020)084]

• Antonio Gallerati, Supersymmetric theories and graphene, in 40th International Conference on High Energy physics (ICHEP2020), PoS 390 (2021) [arXiv:2104.07420, doi:10.22323/1.390.0662]

Exposition:

• Bianca L. Cerchiai, Supergravity in a pencil, talk at M-Theory and Mathematics 2020, NYU Abu Dhabi [pdf slides, video: YT]

• Bianca L. Cerchiai, Holography, Supergravity and Graphene, talk at 106th online SIF Congress (2020) [pdf, pdf&r

category: physics