Analytical Modeling of Graphene Plasmons

Abstract

The two-dimensionality of graphene and other layered materials can be exploited to simplify the theoretical description of their plasmonic and polaritonic modes. We present an analytical theory that allows us to simulate these excitations in terms of plasmon wave functions (PWFs). Closed-form expressions are offered for their associated extinction spectra, involving only two real parameters for each plasmon mode and graphene morphology, which we calculate and tabulate once and for all. Classical and quantum-mechanical formulations of this PWF formalism are introduced, in excellent mutual agreement for armchaired islands with > 10 nm characteristic size. Examples of application are presented to predict both plasmon-induced transparency in interacting nanoribbons and excellent sensing capabilities through the response to the dielectric environment. We argue that the PWF formalism has general applicability and allows us to analytically describe a wide range of 2D polaritonic behavior, thus facilitating their use for the design of actual devices. Plasmons are collective oscillations of conduction electrons found in different materials, where they interact strongly with light and can confine it down to nanoscale spatial regions to generate enormous optical field intensity enhancement [1]. These extraordinary properties are of paramount importance for a wide range of applications, such as optical sensing and modulation [2][3][4][5][6], the enhancement of nonlinear optical processes [7,8], photocatalysis [9][10][11][12][13][14], and photothermal therapies [15,16]. In these applications, precise spectral positioning of plasmon resonances is needed to achieve optimal performance. This is commonly achieved by fabricating noble metal nanostructures with specific sizes and morphologies. However, despite being the workhorse of plasmonics research, noble metals unfortunately present relatively large inelastic losses, thus limiting plasmon lifetimes in metallic nanostructures [17] and leading to a severe reduction in optical confinement. Additionally, the large number of electrons involved in the plasmons of metallic nanostructures limits the ways in which we can influence them in a dynamical fashion.Recently, highly-doped graphene has emerged as an outstanding plasmonic material [18][19][20][21][22][23][24][25][26][27][28][29][30][31] that simultaneously provides strong field confinement with relatively lower loss [32]. More importantly, plasmons in graphene are sustained by a small number of charge carriers compared to those of traditional noble metals, a property that makes them amenable to display new phenomena, including an unprecedented electro-optical response. Indeed, active tunability of the plasmon resonance frequency has been achieved via electrical gating [21][22][23][24][25][26][28][29][30]. Additionally, many of the aforementioned applications that were first realized using noble metal plasmons have now been realized using a tunable graphene platform [31][32][33][34]. However, the design of graphene-based plasmoni...

Strong Light–Matter Interactions Enabled by Polaritons in Atomically Thin Materials

“…Notwithstanding, the vibrant field of graphene plasmonics is considerably wealthier, going well beyond the restricted subgroup of infinite, extended graphene structures. Below, we shall expand our previous considerations by discussing localized plasmons in graphene nanostructures . These are typically fabricated by patterning an otherwise continuous graphene sheet, and arguably constitute the most abundant subset of graphene‐based plasmonic structures considered in the scientific literature.…”

“…Moreover, owing to the extreme subwavelength confinement promoted by GPs, the nonretarded limit (defined by the absence of retardation effects) can be taken without any loss of accuracy in nearly all the relevant scenarios. In fact, treating plasmons in graphene within a nonretarded approach often constitutes an excellent approximation, as demonstrated in a number of works . In this vein, we take the nonretarded limit (i.e., q ≫ k 0 ) of Equation , thereby obtaining the nonretarded dispersion relation of graphene plasmons where we have introduced the quantity without loss of generality .…”

“…In general, this cannot be done analytically and therefore one has to rely on numerical methods. In spite of this, it is still possible to solve Equation using semianalytical techniques . Broadly speaking, the idea behind such semianalytical methods is to expand the electric potential using a suitable set of basis functions.…”

“…The disk's eigenmodes of dipole character, i.e., with l = 1, are of particular importance with respect to their excitation and interaction with plane‐waves, since they can couple strongly to these. Evidently, in the nonretarded limit the fundamental dipole mode {1, 1} dominates the optical response of the system under plane‐wave illumination, and whose absorption cross section has been shown to be quite substantial …”

“…, which constitutes an extension of the electrostatic eigenmode expansion for the linear optical response outlined in ref. (in a way reminiscent of the framework detailed in Section ) to higher‐order processes. In this model (presented in this section in Gaussian units), the response associated with a nonlinear optical process of order n and harmonic s [e.g., n = s = 3 for third‐harmonic generation (THG)] to an external field E ext e −i ωt + c.c.…”