# nLab graphene

Contents

### Context

#### Topological physics

Topological Physics – Phenomena in physics controlled by the topology (often: the homotopy theory) of the physical system.

General theory:

# Contents

## 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).

## References

### General

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.)

Further discussion of the electron band structure of graphene:

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:

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:

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]

### 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:

category: physics

Last revised on September 12, 2022 at 14:32:47. See the history of this page for a list of all contributions to it.