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Nonclassical Graphene Plasmonics

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From Classical to Quantum Plasmonics in Three and Two Dimensions

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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.

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Notes

  1. 1.

    The low-frequency degeneracy pressure of a Dirac system of density \(n=N/A\) is derivable as the (negative) area (A) derivative of the total internal energy \(U \propto N^{3/2}A^{-1/2}\) for fixed particle number N.

  2. 2.

    The previously non-subscripted conductivity \(\sigma =\sigma _{\scriptscriptstyle \text {intra}} + \sigma _{\scriptscriptstyle \text {inter}} \) is here appended with a B-subscript in anticipation of an impending need to differentiate several distinct local conductivity contributions.

  3. 3.

    We remind the low-loss low-temperature intra- and interband conductivities of Eq. (4.12): \(\sigma _{\scriptscriptstyle \text {intra}} (\omega )= \frac{ \mathrm {i} e^2}{\pi \hbar \tilde{\omega } }\) and \(\sigma _{\scriptscriptstyle \text {inter}} (\omega ) = \frac{ \mathrm {i} e^2}{4\pi \hbar } \ln \big |\frac{2- \tilde{\omega } }{2+ \tilde{\omega } }\big |\).

  4. 4.

    The inclusion of loss, absent in the above considerations, can be achieved by straightforward modifications to \(\sigma _{B}(\omega )\) and the substitution \(\beta ^2\!\rightarrow \!\tfrac{\omega }{\omega + \mathrm {i} \gamma }\beta ^2\), see Eq. (3.13) and our earlier treatment of the parabolic HMD.

  5. 5.

    An error exists in the specification of the Green function employed in Ref. [6]. We indicate the correct form here: for width-normalized momenta q and coordinates x, it should read \(G(x,x') = \frac{1}{2q}\Big \lbrace \mathrm {e}^{-q|x-x'|}+\frac{\mathrm {e}^{-q}\cosh [q(x-x')]+\cosh [q(x+x')]}{\sinh q}\Big \rbrace \), as derivable from a minor extension of the half-sheet result [7].

  6. 6.

    Surprisingly, the earliest discussions of edge states in graphene precede its discovery by nearly two decades [11], studied then in the context of stacked benzene-chains.

  7. 7.

    Lieb [13] offered a similar argument even earlier than Inui et al. [12], but did not comment directly on edge localization.

  8. 8.

    The variation of coupling strength for edge-atoms is also ignored in the TB considerations to be discussed shortly—pragmatically, we assume the otherwise dangling bonds passivated by hydrogen atoms, and neglect \(p_z\)-orbital modifications arising therefrom.

  9. 9.

    \(\text {DOS}( {\epsilon } )= \tfrac{2}{\pi \mathcal {A}}\text {Im} \sum _j \frac{1}{ {\epsilon } _{j} - {\epsilon } - \mathrm {i} \hbar \eta }\) for sample area \(\mathcal {A}\) and level-broadening \(\eta \).

  10. 10.

    \(\text {LDOS}( \mathbf {r} _l, {\epsilon } ) = \tfrac{2}{\pi \mathcal {A}}\text {Im} \sum _j \frac{|\psi _{jl}|^2}{ {\epsilon } _{j} - {\epsilon } - \mathrm {i} \hbar \eta }\).

  11. 11.

    An additional class of localized states arise near the van Hove singularity at \(\pm t_{\textsc {ab}}\) (but does not appear in the energy range of Fig. 6.3).

  12. 12.

    Further and much earlier support is found in the 1996 treatment of Ref. [19], in their (numerical) consideration of arbitrarily terminated nanoribbons.

  13. 13.

    For brevity, and since there is no chance of confusion, we henceforth omit explicit declaration of in-plane quantities, i.e. we omit \(\parallel \)-subscripts.

  14. 14.

    The real-space form of \(\chi ^0\) is central to the linear-response real-space and frequency-formulation of TDDFT, sometimes referred to as the Lehmann representation [37], which in allows accounting of the (dynamic) effects of xc-interaction.

  15. 15.

    Equivalently, the RPA density-density function \(\varvec{\chi }^{\textsc {rpa}} = \varvec{\chi }^0(1-e^2\varvec{V}\varvec{\chi }^0)^{-1}=(1-e^2\varvec{\chi }^0 \varvec{V})^{-1}\varvec{\chi }^0\) relates the induced charge density to the external perturbation via \(\varvec{\rho } = e^2\varvec{\chi }^{\textsc {rpa}}\varvec{\phi }^{\scriptscriptstyle \text {ext}} \).

  16. 16.

    We assume a perturbation effective from \(t=0\), such that the ground state density is given by \(n_l(0)\).

  17. 17.

    Strictly speaking, a time-ordering operator \(\hat{\mathcal {T}}\) should be included in the exponential form of \(\hat{U}(t,t')\)—but its absence is of no consequence presently.

  18. 18.

    While this heuristic approach to loss provides an excellent (TDDFT-) account of nonlinearities in e.g. molecules, it is not appropriate to transfer to plasmonic circumstances, where field-enhancement plays a major dynamic role. In our calculations, we use perturbations of sufficient weakness to ensure linearity.

  19. 19.

    All time-domain calculations discussed here assume zero temperature, i.e. \(T=0\).

  20. 20.

    A meaningful assessment of nonclassical shifts require—in addition to precise quantum predictions—a highly accurate classical approach: crucially, the specialized methods discussed in Sect. 5.3.2 achieve this.

  21. 21.

    In the opposite limit of very small sizes, \(\sqrt{\text {area}}\lesssim 8\text { nm}\), the plasmon peak is poorly developed, involving just a few single-particle transitions.

  22. 22.

    Equation (6.13) is strictly speaking a Dirac–Weyl equation, differentiated from the Dirac counterpart by the absence of a mass-term. Nevertheless, the distinction is seldom emphasized for graphene.

  23. 23.

    Their relevant Kronecker products have the explicit forms: \( \tau _{\scriptscriptstyle 0} \otimes \sigma _x \!=\! \left[ {\begin{matrix} 0 &{} 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 &{} 0 \end{matrix}}\right] \) and \( \tau _z\otimes \sigma _y \!=\! \left[ {\begin{matrix} 0 &{} - \mathrm {i} &{} 0 &{} 0 \\ \mathrm {i} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} \mathrm {i} \\ 0 &{} 0 &{} - \mathrm {i} &{} 0 \end{matrix}}\right] \).

  24. 24.

    We note two additional BCs for general domains \(\Omega \) with boundary normal \(\hat{\mathbf {n}} = \big [ {\begin{matrix} \cos \theta \\ \sin \theta \end{matrix}} \big ]\):

    1. a.

      AC terminations admix the valleys at \( \mathbf {r} \in \partial \Omega \): \(\psi _{\textsc {a},\textsc {b}}^+( \mathbf {r} )/\psi _{\textsc {a},\textsc {b}}^-( \mathbf {r} )= -\mathrm {e}^{ \mathrm {i} ( \mathbf {K} ^- - \mathbf {K} ^+)\cdot \mathbf {r} }\) [48].

    2. b.

      Infinite mass confinement—included by a term \( v_{\textsc {f}} ^2 m( \mathbf {r} )\sigma _z\) with \(m( \mathbf {r} ) = \big \lbrace {\begin{matrix} 0, &{} \mathbf {r} \in \Omega \\ \infty , &{} \mathbf {r} \notin \Omega \end{matrix}}\) added into \(\hat{H}_{\textsc {d}}^\kappa \)—decouples the valleys but enforces a sublattice phase relation \(\phi _{\textsc {b}}^\kappa ( \mathbf {r} )/\psi _{\textsc {a}}^\kappa ( \mathbf {r} ) = \mathrm {i} \mathrm {e}^{ \mathrm {i} \kappa \theta }\) at \( \mathbf {r} \in \partial \Omega \) [49].

  25. 25.

    In words, this double series sums all dipole-allowed transitions between filled edge states and empty bulk states.

  26. 26.

    In all cases, level-quantization also plays a role, particularly in the very small size limit, fracturing there the main plasmon peak into several subbands. In the semiclassical limit, it conceivably adds also to the nonlocal blueshift though, we expect, to a lesser degree.

  27. 27.

    The Kerr correction is of a self-focusing type, and accordingly must be augmented to include a saturating mechanism, or suffer runaway focusing (an issue familiar from optical waveguide modeling [75]). We achieve this in practice by using the two-level saturation model consistent with Eq. (6.18) plus a phenomenological accounting of two-photon absorption [73].

  28. 28.

    \(E_{\scriptscriptstyle (3)} \) is defined through a loss-modified frequency \(\tilde{\omega }^2_{\scriptscriptstyle (3)} \equiv (\omega +\tfrac{ \mathrm {i} }{2}\gamma )(\omega - \mathrm {i} \gamma )\). We emphasize that this frequency is not just \((\omega +\tfrac{ \mathrm {i} }{2}\gamma )^2\) as a linear time-relaxation approximation would suggest: this underscores our previous warning that the time-domain TB-RPA approach fails in the nonlinear regime, since it does not provide a dynamic accounting of loss.

  29. 29.

    Equation (6.19) is a special case of a general result derived in Publication A: for a nonperturbed setup \(\{\Omega ,f^{\scriptscriptstyle (0)} \}\) with solutions \(\{\zeta _\nu ^{\scriptscriptstyle (0)} , \mathbf {E} _\nu ^{\scriptscriptstyle (0)} \}\) subjected to a small perturbation \(f = f^{\scriptscriptstyle (0)} + f^{\scriptscriptstyle (1)} \), the perturbed eigenvalues \(\zeta _\nu \simeq \zeta _\nu ^{\scriptscriptstyle (0)} + \zeta _\nu ^{\scriptscriptstyle (1)} + \ldots \) attain a first-order shift \(\zeta _\nu ^{\scriptscriptstyle (1)} \simeq \zeta _\nu ^{\scriptscriptstyle (0)} \big \langle \mathbf {E} _\nu ^{\scriptscriptstyle (0)} \big |f^{\scriptscriptstyle (0)} f^{\scriptscriptstyle (1)} \big | \mathbf {E} _\nu ^{\scriptscriptstyle (0)} \big \rangle \big / \big \langle \mathbf {E} _\nu ^{\scriptscriptstyle (0)} \big |f^{\scriptscriptstyle (0)} \big | \mathbf {E} _\nu ^{\scriptscriptstyle (0)} \big \rangle \).

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    Thomas Christensen

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Christensen, T. (2017). Nonclassical 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_6

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