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Electronic and Optical Properties of Graphene

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Plasmonics and Light–Matter Interactions in Two-Dimensional Materials and in Metal Nanostructures

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Abstract

Graphene has been hailed as a wonder material due to its singular optoelectronic properties. Notably, most of the awe-inspiring electronic properties of graphene can be understood from elementary considerations about its crystal structure and through the application of standard models of bandstructure theory. In this spirit, in this chapter we provide a cursory review of graphene’s bandstructure and its basic electronic properties. These considerations will then supply us with the necessary ingredients for determining the optical response of graphene.

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Notes

  1. 1.

    Specifically, the \(\text{ sp }^2\)-hybridization—occurring in both graphene and graphite—involves the superposition of the \(2\text{ s }\) orbital with two \(2\text{ p }\) orbitals, say, the \(2\text{ p }_x\) and the \(2\text{ p }_y\) states. On the other hand, the chemical bonding in diamond is due \(\text{ sp }^3\)-hybridization where the four chemical bonds are equivalent (this is also the case, for instance, in methane (\(\text{ CH }_4\)), and in graphane (hydrogenated graphene) [7, 8] or fluorographene (fluorinated graphene) [9, 10]).

  2. 2.

    These can be determined by exploiting the relationship \(\mathbf {a}_i \cdot \mathbf {b}_j = 2\pi \delta _{ij}\) [12].

  3. 3.

    Curiously, the first theoretical description of graphene was actually under the tight-binding approximation, developed by Wallace in 1947 [17] in the context of the band theory of graphite, long before the isolation of monolayer graphene in 2004(–2005) [18,19,20,21].

  4. 4.

    The spin degree of freedom has been omitted here since spin-orbit effects are weak in graphene [25, 26].

  5. 5.

    This observation has contributed to the development of \(\mathbf {k} \cdot \mathbf {p}\) method. See, for instance, Ref. [27].

  6. 6.

    While also performing a unitary transformation of the basis to in order to exclude the \(e^{\pm i\pi /6}\) phase factors (for convenience alone).

  7. 7.

    The Dirac equation with vanishing mass is sometimes also known as the Weyl equation or the Dirac–Weyl equation.

  8. 8.

    This can be easily seen by writing the Dirac Hamiltonians in polar form; for instance, for the \({\varvec{K}}\) valley, Eq. (3.10) [or Eq. (3.12a)], transforms to:

    $$\begin{aligned} \varvec{\mathcal {H}}_{\varvec{K}} = \hbar v_F q \begin{bmatrix} 0 &{} e^{-i\theta _{\mathbf {q}}} \\ e^{i\theta _{\mathbf {q}}} &{} 0 \end{bmatrix}. \end{aligned}$$

    .

  9. 9.

    For hole-doped graphene, the expressions presented throughout this section are nevertheless valid upon performing the replacement \(E_F \rightarrow |E_F|\).

  10. 10.

    Recall that absorption is encoded in the real part of the system’s conductivity, Re \(\sigma \).

  11. 11.

    The noninteracting density-density correlation function \(\chi _0(q,\omega )\) also goes by the name of noninteracting (or bare) density-density response function, or polarizability, Lindhard function [36], or even as the bare pair-bubble diagram in the language of Feynman diagrammatics [2, 37, 38].

  12. 12.

    The detailed derivation of the density-density response function for independent electrons is somewhat lengthy and therefore here we simply outline the main steps and results. A thorough derivation of the density-density response function for the noninteracting homogeneous 3D electron gas can be found in a number of textbooks in condensed matter theory, see, for instance, Refs. [37,38,39,40,41]. The version for a homogeneous 2D electron gas with parabolic dispersion is outlined in Refs. [39, 40], whereas the derivation of the same quantity for graphene is presented in Refs. [2, 42, 43].

  13. 13.

    The derivation of this result can be found in Refs. [2, 42, 43], and it relies on the use of standard techniques of complex analysis, such as, for instance, the application of the Sokhotski–Plemelj formula: \(\lim _{\eta \rightarrow 0^+} \int _{-\infty }^{\infty } \frac{f(x)}{x \pm i \eta } dx = \mathcal {P} \int _{-\infty }^{\infty } \frac{f(x)}{x} dx \mp i \pi f(0)\), where \(\mathcal {P}\) denotes the Cauchy principal value.

  14. 14.

    Clearly, in the longwavelength limit, i.e., \(q \rightarrow 0\) (vertical transitions only), the local version of Pauli blocking encountered in Eqs. (3.21) and Fig. 3.4 is reinstated.

  15. 15.

    In this context, besides the works referenced in the text, it is also just and appropriate to mention other seminal contributions made by a number of people during the 1950s, including, for instance, Bohm and Pines [48,49,50,51,52,53], Brout and Sawada [54,55,56], and Nozières and Pines [57,58,59]. Indeed, these works established the existence of collective plasma oscillations due to the long-range Coulomb interaction between electrons in a homogeneous electron gas, and are often considered as the foundational bedrock for the quantum theory of plasmons. The term “plasmon” was introduced by Pines [53] as the quantum of elementary excitation associated with such collective oscillation.

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  1. Center for Nano Optics, University of Southern Denmark, Odense M, Denmark

    Paulo André Dias Gonçalves

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Gonçalves, P.A.D. (2020). Electronic and Optical Properties of Graphene. In: Plasmonics and Light–Matter Interactions in Two-Dimensional Materials and in Metal Nanostructures. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-38291-9_3

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