Lindhard Dielectric Function in the Relaxation-Time Approximation
About: This article is published in Physical Review B.The article was published on 1970-03-01. It has received 813 citations till now. The article focuses on the topics: Cole–Cole equation.
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TL;DR: Graphene plasmons have been proposed as a platform for strongly enhanced light-matter interactions in this paper, where the authors predict unprecedented high decay rates of quantum emitters in the proximity of a carbon sheet, observable vacuum Rabi splittings, and extinction cross sections exceeding the geometrical area in graphene nanoribbons and nanodisks.
Abstract: Graphene plasmons provide a suitable alternative to noble-metal plasmons because they exhibit much tighter confinement and relatively long propagation distances, with the advantage of being highly tunable via electrostatic gating. Here, we propose to use graphene plasmons as a platform for strongly enhanced light–matter interactions. Specifically, we predict unprecedented high decay rates of quantum emitters in the proximity of a carbon sheet, observable vacuum Rabi splittings, and extinction cross sections exceeding the geometrical area in graphene nanoribbons and nanodisks. Our theoretical results provide the basis for the emerging and potentially far-reaching field of graphene plasmonics, offering an ideal platform for cavity quantum electrodynamics, and supporting the possibility of single-molecule, single-plasmon devices.
2,379 citations
TL;DR: In this article, the authors show that plasmons in doped graphene simultaneously enable low-loss and significant wave localization for frequencies below that of the optical phonon branch hbar omega{;Oph};\approx 0.2 eV.
Abstract: We point out that plasmons in doped graphene simultaneously enable low-losses and significant wave localization for frequencies below that of the optical phonon branch hbar omega_{;Oph};\approx 0.2 eV. Large plasmon losses occur in the interband regime (via excitation of electron-hole pairs), which can be pushed towards higher frequencies for higher doping values. For sufficiently large dopings, there is a bandwidth of frequencies from omega_{;Oph}; up to the interband threshold, where a plasmon decay channel via emission of an optical phonon together with an electron-hole pair is nonegligible. The calculation of losses is performed within the framework of a random-phase approximation and number conserving relaxation-time approximation. The measured DC relaxation-time serves as an input parameter characterizing collisions with impurities, whereas the contribution from optical phonons is estimated from the influence of the electron-phonon coupling on the optical conductivity. Optical properties of plasmons in graphene are in many relevant aspects similar to optical properties of surface plasmons propagating on dielectric-metal interface, which have been drawing a lot of interest lately because of their importance for nanophotonics. Therefore, the fact that plasmons in graphene could have low losses for certain frequencies makes them potentially interesting for nanophotonic applications.
1,983 citations
Cites background or methods from "Lindhard Dielectric Function in the..."
...To find dispersion of plasmons in graphene we need the conductivity of graphene σ(ω, q), which we now proceed to analyze by employing the semiclassical model [33] (in subsection III A), RPA and number conserving relaxation-time approximation [29] (in subsection III B), and by estimating the relaxation-time due to the influence of electron-phonon coupling [30] on the optical conductivity [31] (in subsection III C)....
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...[29] N.D. Mermin, Lindhard Dielectric Function in the Relaxation-Time Approximation, Phys....
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...In order to take the interband losses into account, we use the self-consistent linear response theory, also known as the random-phase approximation (RPA) [33], together with the relaxation-time (finite τ) approximation introduced by Mermin [29]....
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...The effects of other types of scattering (impurities, phonons) can be accounted for by using the relaxation-time τ as a parameter within the RPA-RT approach [29], which takes into account conservation of local electron number....
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...[33] N.W. Ashcroft, N.D. Mermin, Solid State Physics, (Saunders, Philadelphia, PA, 1976)....
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TL;DR: In this paper, a quantum-mechanical description of the interaction between the electrons and the sample is discussed, followed by a powerful classical dielectric approach that can be in practice applied to more complex systems.
Abstract: This review discusses how low-energy, valence excitations created by swift electrons can render information on the optical response of structured materials with unmatched spatial resolution. Electron microscopes are capable of focusing electron beams on sub-nanometer spots and probing the target response either by analyzing electron energy losses or by detecting emitted radiation. Theoretical frameworks suited to calculate the probability of energy loss and light emission (cathodoluminescence) are revisited and compared with experimental results. More precisely, a quantum-mechanical description of the interaction between the electrons and the sample is discussed, followed by a powerful classical dielectric approach that can be in practice applied to more complex systems. We assess the conditions under which classical and quantum-mechanical formulations are equivalent. The excitation of collective modes such as plasmons is studied in bulk materials, planar surfaces, and nanoparticles. Light emission induced by the electrons is shown to constitute an excellent probe of plasmons, combining sub-nanometer resolution in the position of the electron beam with nanometer resolution in the emitted wavelength. Both electron energy-loss and cathodoluminescence spectroscopies performed in a scanning mode of operation yield snap shots of plasmon modes in nanostructures with fine spatial detail as compared to other existing imaging techniques, thus providing an ideal tool for nanophotonics studies.
1,288 citations
TL;DR: Graphene plasmons are rapidly emerging as a viable tool for fast electrical manipulation of light as mentioned in this paper, and the prospects for applications to electro-optical modulation, optical sensing, quantum plasmonics, light harvesting, spectral photometry, and tunable lighting at the nanoscale are further stimulated by the relatively low level of losses and high degree of spatial confinement that characterize these excitations compared with conventional plasmic materials.
Abstract: Graphene plasmons are rapidly emerging as a viable tool for fast electrical manipulation of light. The prospects for applications to electro-optical modulation, optical sensing, quantum plasmonics, light harvesting, spectral photometry, and tunable lighting at the nanoscale are further stimulated by the relatively low level of losses and high degree of spatial confinement that characterize these excitations compared with conventional plasmonic materials. We start with a general description of the plasmons in extended graphene, followed by analytical methods that lead to reasonably accurate estimates of both the plasmon energies and the strengths of coupling to external light in graphene nanostructures, including graphene ribbons. We discuss several possible strategies to extend these plasmons towards the visible and near infrared, including a reduction in the size of the graphene structures and an increase in the level of doping. Specifically, we discuss plasmons in narrow ribbons and molecular-size graphene structures. We further formulate prescriptions based on geometry to increase the level of electrostatic doping without causing electrical breakdown. Results are also presented for plasmons in highly-doped single-wall carbon nanotubes, which exhibit similar characteristics as narrow ribbons and show a relatively small dependence on the chirality of the tubes. We further discuss perfect light absorption by a single-atom carbon layer, which we illustrate by investigating arrays of ribbons using fully analytical expressions. Finally, we explore the possibility of exploiting optically pumped transient plasmons in graphene, whereby the optically heated graphene valence band can sustain collective plasmon oscillations similar to those of highly doped graphene, and well-defined during the picosecond time window over which the electron is at an elevated temperature.
900 citations
TL;DR: In this article, a general description of the plasmonic behavior of extended graphene is given, followed by analytical methods that lead to reasonably accurate estimates of both the energy and the strength of coupling to external light in graphene nanostructures.
Abstract: Graphene plasmons are rapidly emerging as a viable tool for fast electrical manipulation of light. The prospects for applications to electro-optical modulation, optical sensing, quantum plasmonics, light harvesting, spectral photometry, and tunable lighting at the nanoscale are further stimulated by the relatively low level of losses and high degree of spatial confinement that characterize these excitations compared with conventional plasmonic materials, alongside the large nonlinear response of graphene. We start with a general description of the plasmonic behavior of extended graphene, followed by analytical methods that lead to reasonably accurate estimates of both the plasmon energies and the strengths of coupling to external light in graphene nanostructures, including graphene ribbons. Although graphene plasmons have so far been observed at mid-infrared and longer wavelengths, there are several possible strategies to extend them toward the visible and near-infrared, including a reduction in the size ...
828 citations