Skip to main content
SpringerLink
Log in

Electronic Properties of Graphene

  • Chapter
  • First Online:
From Classical to Quantum Plasmonics in Three and Two Dimensions

Part of the book series: Springer Theses ((Springer Theses))

Abstract

We first review the necessary components of electrodynamics and establish a practical notion of optical excitations. Next, we specialize to plasmonic excitations, and discuss their classical features. Finally, we cover theoretical aspects of techniques that probe the properties of plasmons in practice.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Although the Nobel-awarded 2004 paper [9] often is juxtaposed with the “discovery” of graphene, its introduction was far more gradual, and indeed older, than that [10, 11], see e.g. the comments of Refs. [12] and [13].

  2. 2.

    All vectorial quantities in this chapter are implicitly in-plane, i.e. two-dimensional; in Chap. 5 we introduce additional notation to distinguish vectors of different dimensionality.

  3. 3.

    The \(\mathbf {M}\)-point coincides with a van Hove singularity; its importance, however, is minor for our low-energy considerations.

  4. 4.

    Different values of the hopping parameter proliferate: we adopt the value of the prevailing theoretical review [8].

  5. 5.

    Jointly, these properties earn graphene the classification of ‘semi-metal’.

  6. 6.

    We mention one minor modification, namely the opening of a very small bandgap \(\sim \! 1\ \mu \,\text {eV}\) when spin-orbit interaction is included [23].

  7. 7.

    Though we here make a distinction between \( \mathbf {k} \) and \( \mathbf {q} \), we will generally prefer the Dirac perspective; accordingly, both \( \mathbf {k} \) and \( \mathbf {q} \) are henceforth implicitly measured relative to a Dirac point.

  8. 8.

    We warn in advance, that the Dirac equation studied in Sect. 6.2.3 does not follow directly from our present considerations, corresponding instead to a rotated and reflected lattice, preferred for reasons of algebraic simplicity.

  9. 9.

    Six Dirac points reside on the boundary of the FBZ, each including a third of their vicinity inside the FBZ: thus, the valley degeneracy sums to \(6\times \tfrac{1}{3}=2\).

  10. 10.

    Even for thin-films, e.g. a \(10\times 10\times 0.3 \,\text {nm}^3\) gold film, one requires \(\sim \!50\) electrons for a \(0.1 \,\text {eV}\) Fermi energy shift.

  11. 11.

    Throughout this section we assume \( {\epsilon } _{\textsc {f}} >0\); hole-doped equivalents follow by replacing \( {\epsilon } _{\textsc {f}} \rightarrow | {\epsilon } _{\textsc {f}} |\).

  12. 12.

    Mermin’s critical insight was the observation that scattering incurs relaxation not to a global equilibrium but to a local equilibrium which accounts for the influence of a nonzero perturbing field.

  13. 13.

    The link ignores retardation effects since \(\chi ^0(q,\omega )\) is a nonretarded construct; as a consequence it does not distinguish between transverse and longitudinal perturbations.

  14. 14.

    The intraband term’s small temperature dependence can be appreciated from the \(x=T_{\textsc {f}}/T\ll 1\) expansion \(\ln (2\cosh \tfrac{1}{2}x)\simeq \tfrac{1}{2}x + \mathrm {e}^{-x} +\ldots \), revealing [when comparing with Eq. (4.11a)] that the finite-temperature corrections to Eq. (4.12a) are exponentially damped.

  15. 15.

    The relation is obtained by comparing the dc-conductivity definition of the mobility, \(\sigma =n_0 e\mu _{\scriptscriptstyle \text {e}} \), with the low-frequency intraband Drude conductivity \(\sigma (\omega \rightarrow 0) = e^2 {\epsilon } _{\textsc {f}} /\pi \hbar ^2\gamma \).

References

  1. A.N. Grigorenko, M. Polini, K.S. Novoselov, Graphene plasmonics. Nat. Photonics 6, 749 (2012)

    Article  ADS  Google Scholar 

  2. M. Jablan, M. Soljačić, H. Buljan, Plasmons in graphene: fundamental properties and potential applications. Proc. IEEE 101, 1689 (2013)

    Article  Google Scholar 

  3. Y.V. Bludov, A. Ferreira, N.M.R. Peres, M.I. Vasilevskiy, A primer on surface plasmon-polaritons in graphene. Int. J. Mod. Phys. B 27, 1341001 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. F. Xia, H. Wang, D. Xiao, M. Dubey, A. Ramasubramaniam, Two-dimensional material nanophotonics. Nat. Photonics 8, 899 (2014)

    Article  ADS  Google Scholar 

  5. F.J. García de Abajo, Graphene plasmonics: challenges and opportunities. ACS Photonics 1, 135 (2014)

    Article  Google Scholar 

  6. T. Low, P. Avouris, Graphene plasmonics for terahertz to mid-infrared applications. ACS Nano 8, 1086 (2014)

    Article  Google Scholar 

  7. T. Stauber, Plasmonics in Dirac systems: from graphene to topological insulators. J. Phys.: Condens. Matter 26, 123201 (2014)

    Google Scholar 

  8. A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81, 109 (2009)

    Google Scholar 

  9. K.S. Novoselov, A.K. 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, 666 (2004)

    Article  ADS  Google Scholar 

  10. H.P. Boehm, A. Clauss, G.O. Fischer, U. Hofmann, The absorption properties of very thin carbon films, Z. Anorg. Allg. Chem. 316, 119 (1962), originally ‘Das Adsorptionsverhalten sehr dünner Kohlenstoff-Folien’

    Google Scholar 

  11. N.R. Gall, E.V. Rut’kov, A.Y. Tontegode, Two dimensional graphite films on metal and their intercalation. Int. J. Mod. Phys. B 11, 1865 (1997)

    Article  ADS  Google Scholar 

  12. E.S. Reich, Nobel prize committee under fire, Nature News (2010)

    Google Scholar 

  13. W. de Heer, Letter to the 2010 Nobel committee, online (2010)

    Google Scholar 

  14. K.S. Novoselov, A.K. 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 (2005)

    Article  ADS  Google Scholar 

  15. K.S. Novoselov, D. Jiang, F. Schedin, T.J. Booth, V.V. Khotkevich, S.V. Morozov, A.K. Geim, Two-dimensional atomic crystals. Proc. Natl. Acad. Sci. U.S.A. 102, 10451 (2005)

    Article  ADS  Google Scholar 

  16. P.R. Wallace, The band theory of graphite. Phys. Rev. 71, 622 (1947)

    Article  ADS  MATH  Google Scholar 

  17. J.C. Slonczewski, P.R. Weiss, Band structure of graphite. Phys. Rev. 109, 272 (1958)

    Google Scholar 

  18. G.W. Semenoff, Condensed-matter simulation of a three-dimensional anomaly. Phys. Rev. Lett. 53, 2449 (1984)

    Article  ADS  Google Scholar 

  19. K.W.-K. Shung, Dielectric function and plasmon structure of stage-1 intercalated graphite. Phys. Rev. B 34, 979 (1986)

    Article  ADS  Google Scholar 

  20. R. Saito, M. Fujita, G. Dresselhaus, M.S. Dresselhaus, Electronic structure of chiral graphene tubules. Appl. Phys. Lett. 60, 2204 (1992)

    Article  ADS  Google Scholar 

  21. K. Nakada, M. Fujita, G. Dresselhaus, M.S. Dresselhaus, Edge state in grapheme ribbons: nanometer size effect and edge shape dependence. Phys. Rev. B 54, 17954 (1996)

    Article  ADS  Google Scholar 

  22. S. Reich, J. Maultzsch, C. Thomsen, P. Ordejón, Tight-binding description of graphene. Phys. Rev. B 66, 035412 (2002)

    Article  ADS  Google Scholar 

  23. Y. Yao, F. Ye, X.L. Qi, S.C. Zhang, Z. Fang, Spin-orbit gap of graphene: first-principles calculations. Phys. Rev. B 75, 041401 (2007)

    Article  ADS  Google Scholar 

  24. H. Liu, Y. Liu, D. Zhu, Chemical doping of graphene. J. Mater. Chem. 21, 3335 (2011)

    Article  Google Scholar 

  25. C.-F. Chen, C.-H. Park, B.W. Boudouris, J. Horng, B. Geng, C. Girit, A. Zettl, M.F. Crommie, R.A. Segalman, S.G. Louie, F. Wang, Controlling inelastic light scattering quantum pathways in graphene. Nature 471, 617 (2011)

    Article  ADS  Google Scholar 

  26. D.K. Efetov, P. Kim, Controlling electron-phonon interactions in graphene at ultrahigh carrier densities. Phys. Rev. Lett. 105, 256805 (2010)

    Article  ADS  Google Scholar 

  27. J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A.Z. Elorza, N. Camara, F.J. García de Abajo, R. Hillenbrand, F.H.L. Koppens, Optical nano-imaging of gate-tunable graphene plasmons. Nature 487, 77 (2012)

    ADS  Google Scholar 

  28. Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P.M. Ajayan, P. Nordlander, N.J. Halas, F.J. García de Abajo, Gated tunability and hybridization of localized plasmons in nanostructured graphene. ACS Nano 7, 2388 (2013)

    Article  Google Scholar 

  29. X. Zhu, W. Wang, W. Yan, M.B. Larsen, P. Bøggild, T.G. Pedersen, S. Xiao, J. Zi, N.A. Mortensen, Plasmon-phonon coupling in large-area graphene dot and antidot arrays fabricated by nanosphere lithography. Nano Lett. 14, 2907 (2014)

    Article  ADS  Google Scholar 

  30. Z. Fang, Y. Wang, A.E. Schlather, Z. Liu, P.M. Ajayan, F.J. García de Abajo, P. Nordlander, X. Zhu, N.J. Halas, Active tunable absorption enhancement with graphene nanodisk arrays. Nano Letters 14, 299 (2014)

    Article  ADS  Google Scholar 

  31. H. Ehrenreich, M.H. Cohen, Self-consistent field approach to the many-electron problem. Phys. Rev. 115, 786 (1959)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. J. Lindhard, On the properties of a gas of charged particles. Dan. Mat. Fys. Medd. 28, 8 (1954)

    MathSciNet  MATH  Google Scholar 

  33. B. Wunsch, T. Stauber, F. Sols, F. Guinea, Dynamical polarization of grapheme at finite doping. New J. Phys. 8, 318 (2006)

    Article  ADS  Google Scholar 

  34. E.H. Hwang, S. Das, Sarma, Dielectric function, screening, and plasmons in two-dimensional graphene. Phys. Rev. B 75, 205418 (2007)

    Article  ADS  Google Scholar 

  35. N.D. Mermin, Lindhard dielectric function in the relaxation-time approximation. Phys. Rev. B 1, 2362 (1970)

    Article  ADS  Google Scholar 

  36. B.E. Sernelius, Retarded interactions in graphene systems. Phys. Rev. B 85, 195427 (2012)

    Article  ADS  Google Scholar 

  37. H. Bruus, K. Flensberg, Many-Body Quantum Theory in Condensed Matter Physics (Oxford University Press, 2004)

    Google Scholar 

  38. L.A. Falkovsky, A.A. Varlamov, Space-time dispersion of graphene conductivity. Eur. Phys. J. B 56, 281 (2007)

    Article  ADS  Google Scholar 

  39. F.H.L. Koppens, D.E. Chang, F.J. García de Abajo, Graphene plasmonics: a platform for strong light-matter interactions. Nano Lett. 11, 3370 (2011)

    Article  ADS  Google Scholar 

  40. G.W. Hanson, Quasi-transverse electromagnetic modes supported by a grapheme parallel-plate waveguide. J. Appl. Phys. 104, 084314 (2008)

    Article  ADS  Google Scholar 

  41. P. Tassin, T. Koschny, C.M. Soukoulis, Graphene for terahertz applications. Science 341, 620 (2013)

    Article  ADS  Google Scholar 

  42. Z. Fei, G.O. Andreev, W. Bao, L.M. Zhang, A.S. McLeod, C. Wang, M.K. Stewart, Z. Zhao, G. Dominguez, M. Thiemens, M.M. Fogler, M.J. Tauber, A.H. Castro, Neto, C.N. Lau, F. Keilmann, D.N. Basov, Infrared nanoscopy of Dirac plasmons at the graphene \(\rm SiO_{2}\) interface. Nano Lett. 11, 4701 (2011)

    Google Scholar 

  43. H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea, P. Avouris, F. Xia, Damping pathways of mid-infrared plasmons in graphene nanostructures. Nat. Photonics 7, 394 (2013)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, Massachusetts Institute of Technology, Cambridge, USA

    Thomas Christensen

Authors
  1. Thomas Christensen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas Christensen .

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Christensen, T. (2017). Electronic Properties of Graphene. In: From Classical to Quantum Plasmonics in Three and Two Dimensions. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-48562-1_4

Download citation

Publish with us

Policies and ethics

Search

Navigation