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Classical Graphene Plasmonics

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From Classical to Quantum Plasmonics in Three and Two Dimensions

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Abstract

We open this chapter with a general consideration of the role of dimensionality in plasmonics. Subsequently, specializing in earnest to graphene, we examine the existence of plasmons in extended sheets of doped graphene. Finally, following an extensive discussion of plasmons in nanostructured graphene, we explore plasmons in curved graphene, concretely in graphene-coated nanospheres.

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Notes

  1. 1.

    Graphene plasmonics, with very few exceptions, require nonzero doping—the premise of our investigation consequently relies on nonzero doping, i.e. \( {\epsilon } _{\textsc {f}} \ne 0\).

  2. 2.

    E.g., in a recent review of two-dimensional nanophotonics [2], just two citations precede the 2004 discovery of graphene, both in reference to bulk properties.

  3. 3.

    See similar treatments in Refs. [12] and [13].

  4. 4.

    The 1D transform can be performed using \(\int _{-\infty }^{\infty } \frac{\mathrm {e}^{- \mathrm {i} q x}}{\sqrt{x^2 + a^2}} \, \mathrm {d} {x}=2K_0(qa)\). In polar coordinates the 2D transform requires \(\int _0^{2\pi } \mathrm {e}^{- \mathrm {i} x\cos \theta } \, \mathrm {d} \theta = 2\pi J_0(x)\) and \(\int _0^\infty J_0(qr) \, \mathrm {d} {r} = q^{-1}\). Finally, by considering a Yukawa potential \(\mathrm {e}^{-q_{\textsc {y}}r}/r\) in the \(q_{\textsc {y}}\rightarrow 0^+\) limit, the 3D transform can be deduced in spherical coordinates using \(\int _0^\pi \mathrm {e}^{- \mathrm {i} x\cos \theta }\sin \theta \, \mathrm {d} \theta = 2\,\mathrm {sinc}\,x\) and \(\int _0^\infty \mathrm {e}^{-q_{\textsc {y}}r}\sin qr \, \mathrm {d} {r} = \frac{q}{q^2+q_{\textsc {y}}^2}\).

  5. 5.

    The surface-current approach is analogous to the dipole approximation familiar from field-emitter interactions; its validity is guaranteed by the complete fulfillment of the condition \(h_{\scriptscriptstyle \text {g}} /\lambda \ll 1\) for wavelengths \(\lambda \) up to the ultraviolet.

  6. 6.

    More generally, the effective dielectric cladding imparts a rescaling of the in-plane Coulomb interaction \(V( \mathbf {r}_{{\scriptscriptstyle \parallel }} , \mathbf {r}_{{\scriptscriptstyle \parallel }} ') \rightarrow \bar{\varepsilon }^{-1}V( \mathbf {r}_{{\scriptscriptstyle \parallel }} , \mathbf {r}_{{\scriptscriptstyle \parallel }} ')\) in the nonretarded limit. This rescaling is rigorous; we revisit it in Sect. 5.3.

  7. 7.

    The opposite limit, i.e. \(\hbar \omega \ll \alpha {\epsilon } _{\textsc {f}} \), though of little technological or plasmonic importance is noteworthy at least for the sake of completeness. There, as was the case also for metal SPPs, the graphene SPP eventually exhibits predominately polaritonic properties, with dispersion \( k_{\scriptscriptstyle \parallel } \simeq k_0\sqrt{\bar{\varepsilon }} + \mathcal {O}[(\hbar \omega /\alpha {\epsilon } _{\textsc {f}} )^2]\). In other words, the dispersion is ultimately linear for \(\omega \rightarrow 0\).

  8. 8.

    For nonvanishing but small loss, the plasmon frequency \(\omega _{\textsc {gp}}^{\scriptscriptstyle \text {intra}} \) acquires a finite imaginary part \(\simeq \mathrm {i} \gamma /2\).

  9. 9.

    In practice, the dispersion is obtained by minimizing \(|1-V( k_{\scriptscriptstyle \parallel } )\text {Re}\chi ^0( k_{\scriptscriptstyle \parallel } ,\omega )|\) over real, positive \(\{ k_{\scriptscriptstyle \parallel } ,\omega \}\).

  10. 10.

    To include loss it is necessary to solve \(1-V( k_{\scriptscriptstyle \parallel } )\chi ^0( k_{\scriptscriptstyle \parallel } ,\omega )=0\) in \(\{ k_{\scriptscriptstyle \parallel } ,\omega \}\) with either \( k_{\scriptscriptstyle \parallel } \) or \(\omega \) complex, rather than \(1-V( k_{\scriptscriptstyle \parallel } )\text {Re}\chi ^0( k_{\scriptscriptstyle \parallel } ,\omega )=0\) in real variables. Beyond a perturbative approach [19], which is inapplicable in Landau regions, this has not been achieved in nonlocal treatments. Arguably, a quantitative resolution of this issue is of modest practical worth: plasmon properties are certainly poor in these regions.

  11. 11.

    See Appendix A for a treatment of the interaction between a normally incident electron and graphene.

  12. 12.

    We note that the application of the linear, no-recoil EELS framework, Eq. (2.26), to such low acceleration energies carries an undeniable degree of unease: as the energy-loss eventually constitutes a sizable fraction of the total electron energy, the electron is deflected from its straight path, inducing further complication of a self-consistent kind that is not captured by Eq. (2.26).

  13. 13.

    The TE GPP is also limited from above to energies \(\hbar \omega / {\epsilon } _{\textsc {f}} <2\) due to the onset of vertical Landau damping.

  14. 14.

    It was recently suggested, perhaps optimistically, that this restriction on \(\varepsilon -1\) might be turned to functionality in the context of ultra-sensitive sensing of dielectric environments [26].

  15. 15.

    The integration domain \(\tilde{\Omega }\) is indicated in Eq. (5.14a) for mnemonic reasons only: in principle the integral extends over all of \(\tilde{ \mathbf {r} }_{{\scriptscriptstyle \parallel }}'\in \mathbb {R}^2\), but is limited in practice by the extent of \(f(\tilde{ \mathbf {r} }_{{\scriptscriptstyle \parallel }}')\), assumed bounded by \(\tilde{\Omega }\).

  16. 16.

    The gradient of the indicator function is the analogue of the derivative of the Heaviside step function, in the sense that: \(\int _{\scriptscriptstyle \Omega } [\nabla {\mathbbm {1}}_{\scriptscriptstyle \Omega } ( \mathbf {r} )] \cdot \mathbf {g}( \mathbf {r} ) \, \mathrm {d} ^d \mathbf {r} = - \oint _{\scriptscriptstyle \partial \Omega } \hat{\mathbf {n}}\cdot \mathbf {g}( \mathbf {r} ) \, \mathrm {d} ^{d-1} \mathbf {r} \) for \(\Omega \in \mathbb {R}^n\) and outward normal vector \(\hat{\mathbf {n}}\).

  17. 17.

    The surface integral over \(\Omega _+\) reduces to an integral over \(\Omega \) since \({\mathbbm {1}}_{\Omega }( \mathbf {r}_{{\scriptscriptstyle \parallel }} )\) vanishes for \( \mathbf {r}_{{\scriptscriptstyle \parallel }} \!\notin \!\Omega \), i.e. there is no contribution from the small annulus \(\Omega _+\backslash \Omega \).

  18. 18.

    We emphasize that this step does not make use of the BC—it is merely a consequence of the assumed reduction of integration domain from \(\Omega \) to \(\Omega _-\).

  19. 19.

    The response-magnitude at each eigenfrequency will, of course, depend on the associated eigenstates; but is crucially similarly separable in geometric, scale, and material dependencies.

  20. 20.

    We suggest that the comparative absence of analytical solutions relative to 3D metal plasmonics, can be traced to a dimensional mismatch between structure and space: a 2D graphene sample exists in a 3D electromagnetic reality; accordingly, severe restrictions on “good” symmetries are imposed from the outset.

  21. 21.

    We note the relevant definite integrals [29]: \(\int _0^{2\pi } \mathrm {e}^{ \mathrm {i} r\cos \theta } \, \mathrm {d} \theta = 2\pi J_0(r)\) and \(\int _0^\infty J_0(r) \, \mathrm {d} {r} = 1\).

  22. 22.

    The analytical origin of the value \(\zeta _{\scriptscriptstyle \text {edge}} \approx 0.8216\) is provided by the Wiener–Hopf technique: it is the solution to the integral equation \(\int _0^{\pi /2} \ln [(\zeta \sin {x})^{-1}-1] \, \mathrm {d} {x} = 0\) [27].

  23. 23.

    The fits have a mean relative deviation of less than \(\approx \!2\) ‰ compared to the full numerical solution. The monopole fit, \(n=0\), however, is less accurate (\({\approx }1\) %). This is because the monopole is physically distinct from the \(n\ne 0\) modes for \( k_{\scriptscriptstyle \parallel } W \ll 1\) where it exhibits distinctively 1D behavior (see Table 5.1) of the sort \(\zeta _0 \propto ( k_{\scriptscriptstyle \parallel } W)^2 \ln ( k_{\scriptscriptstyle \parallel } W)\), with an associated frequency dispersion \(\omega \propto k_{\scriptscriptstyle \parallel } \sqrt{\ln ( k_{\scriptscriptstyle \parallel } W)}\). An analytical demonstration of this latter point is provided by the variational treatment of Ref. [34].

  24. 24.

    We note an unfortunate typographic error in the Supplementary Material of Publication E: in Eq. (S13c), the term \(\propto \!\delta _{j,k+1}\) is divided by \(8\prod _{p=1}^{3}(l+2j+p)\). This divisor should have read \(8\prod _{p=1}^{3}(l+2k+p)\) in order to provide a symmetric matrix.

  25. 25.

    For an arbitrarily polarized excitation wave, e.g. along \(\hat{\mathbf {n}}_ \mathbf {E} \), the relevant polarizability is trivially obtained from (5.22) by the substitution \(\tilde{x}\rightarrow \hat{\mathbf {n}}_ \mathbf {E} \cdot \tilde{ \mathbf {r} }_{{\scriptscriptstyle \parallel }}\).

  26. 26.

    We remind that the nanoribbon’s polarizability requires special interpretation cf. its semi-infinite extent; nevertheless the concept is still fruitful, e.g. in consideration of absorption per unit length.

  27. 27.

    \(N\times N\) matrix inversion exhibits computational complexity \(\mathcal {O}(N^3)\) for the conventional Gauss-Jordan algorithm; improved scaling can be obtained for large N using iterative methods.

  28. 28.

    The otherwise interesting topic of lattice-coupling, i.e. nanostructures in periodic arrays, we leave entirely aside, although it is straightforwardly treatable by lattice-summation in the dipole limit [47].

  29. 29.

    Graphene bowties, first treated in Ref. [48], is discussed further in Publication C.

  30. 30.

    Although fullerenes represent a tempting small-scale analogy, they are likely not well-described by the theory developed here: in very small fullerenes quantum effects are important and an extended-graphene description accordingly poor, while the larger fullerenes exhibit icosadhedral rather than spherical configurations [63, 64].

  31. 31.

    Several additional examples and explicit calculations related to coated nanospheres are presented in Publication D, including treatments of size-dispersion, far-field extinction, hybridization with an underlying Drude-sphere, and inclusion of hydrodynamic response.

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    Thomas Christensen

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Christensen, T. (2017). Classical Graphene Plasmonics. 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_5

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