Abstract
Many of the fundamental electronic properties of the solid state can be described by the concept of single electrons moving between an ion lattice . If we ignore the lattice, in a first approximation, we end up with a different approach where the free electrons of a metal can be treated as an electron liquid of high density [1, 2] . From this plasma model it follows that longitudinal density fluctuations, so-called plasma oscillations or Langmuir waves , with an energy of the order of 10 eV will propagate through the volume of the metal. These volume excitations have been studied in detail with Electron Energy Loss Spectroscopy (EELS)1 and have led to the discovery of surface plasmon polaritons.
The world is full of magical things patiently waiting for our wits to grow sharper.
Bertrand Russell
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Notes
- 1.
A historical overview of electron beam experiments to study surface plasmons can be found in [3], p. 47.
- 2.
Born 2nd May 1868 in Concord, Massachusetts; † 11th August 1955 in Amityville, New York.
- 3.
Born 12th November 1842 in Langford Grove, Essex; † 30th June 1919 in Witham, Essex. Nobel Prize in Physics 1904.
- 4.
Born 28th July 1912 in Turin; † 13th February 2001 in Chicago.
- 5.
- 6.
Born 12th July 1863 in Braunschweig; † 5th July 1906 in Berlin.
- 7.
Born 18th July 1853 in Arnheim; † 4th February 1928 in Haarlem. Nobel Prize in Physics 1902.
- 8.
Born 5th December 1868 in Königsberg; † 26th April 1951 in München.
- 9.
Born 2th July 1906 in Straßburg; † 6th March 2005 in Ithaca, New York. Nobel Prize in Physics 1967.
- 10.
For gold at room temperature we have ne \(= 3/[4\pi (0.159 \times 10^{-9})^{3}] = 5.9 \times 10^{28}\;\mbox{ electrons}/\mbox{ m}^{3}\). With e \(= 1.602 \times 10^{-19}\) As, \(\varepsilon _{0} = 8.854 \times 10^{-12}\;\mbox{ A$^{2}$ s$^{4}$}/\mbox{ kg m$^{3}$}\), and me \(= 9.109 \times 10^{-31}\;\mbox{ kg}\) we obtain ω p = 13. 75 PHz or ℏ ω p = 9. 05 eV, respectively.
- 11.
In general n \(= \pm \sqrt{\mbox{ $\mu \varepsilon $} /\mbox{ $\mu 0 \varepsilon 0$}} \approx \pm \sqrt{\mbox{ $\varepsilon $} /\mbox{ $\varepsilon 0$}}\) for optical frequencies and the positive sign is chosen for causality reasons in the system. A negative refractive index does not occur in nature but can be artificially generated with metamaterials (see Sect. 9 or e.g. [24–26]).
- 12.
In the literature most of the time the complex refractive index is tabulated, the connection to the dielectric function is given by \(\mbox{ $\varepsilon $}/\mbox{ $\varepsilon 0$} = \mbox{ $\varepsilon $}_{1} +\mathrm{ i}\mbox{ $\varepsilon $}_{2} =\) n \(^{2} = (\mbox{ $\tilde{n}$} +\mathrm{ i}\mbox{ $\tilde{k}$})^{2}\). The real and imaginary parts of \(\mbox{ $\varepsilon $}/\mbox{ $\varepsilon _{0}$}\) then follow as \(\varepsilon _{1} = \mbox{ $\tilde{n}$}^{2} -\mbox{ $\tilde{k}$}^{2}\) and ɛ 2 = 2˜n˜k.
- 13.
The conversion factor between eV and nm is 1239.84 as discussed in Appendix A.1.
- 14.
The abbreviations (s) and (p) come from the German words senkrecht (perpendicular) and parallel (parallel), respectively.
- 15.
- 16.
Typically ζ m is one order of magnitude smaller than ζ b , e.g. for gold we obtain ζ m ≈ 30 nm and ζ b ≈ 280 nm at λ = 600 nm.
- 17.
Since we solely discuss particle plasmon polaritons in this book, we will henceforth always refer to them.
- 18.
The windows of Sainte-Chapelle in Paris are a very good example of this: Light transmission through the metal ions in the stained glass strongly depends on the incident and viewing angles. At sunset, the grazing-angle scattering of light by gold particles in the window creates a pronounced red glow that appears to slowly move downward, while intensities of blue tints from ions of copper or cobalt remain the same [39, 40].
- 19.
Lycurgus cup, fourth century AD, British Museum, London.
- 20.
A unified treatment of fluorescence and Raman scattering can be found in [56] for example.
- 21.
Also an enhancement in polarizability due to chemical effects such as charge-transfer excited states can contribute to the huge SERS signal [55], but in the following we will just stick to the electromagnetic enhancement.
- 22.
- 23.
- 24.
The formation of two hybridized modes (cf. bonding and antibonding modes in Fig. 2.26) oscillating at different energies is a typical indicator of strong coupling, if the involved damping is small. In the time domain the energy then oscillates between atom and cavity (Rabi oscillations, named after the Nobel laureate Isidor Isaac Rabi ) [91]. In the frequency domain we obtain Rabi splitting and anticrossing , see [88]. Rabi splitting is fundamental for the dynamics of two-state systems and can be easily modeled using the Jaynes-Cummings model [92] for example.
- 25.
Fano interference occurs, when a resonant or discrete state interacts with a continuum of states–a very general effect that can be found in many different areas of physics and has been derived originally by Ugo Fano [93].
- 26.
Since the heat capacity of the electronic system is much smaller than that of the ion lattice, an excitation by femtosecond laser pulses can generate extremely high electron temperatures [36].
- 27.
At the beginning of this section we have seen that d-band absorption for gold starts around 620 nm, in silver around 400 nm.
- 28.
This fundamental constant can be interpreted in several ways, e.g. as the electromagnetic coupling strength for the interaction between electrons and photons, as a ratio of charges, energies, or characteristic lengths. When Arnold Sommerfeld introduced this dimensionless number in 1916 to explain the splitting or fine structure of the energy levels of the hydrogen atom, which had been observed, he considered the ratio of the velocity of the electron in the first circular orbit of the Bohr model to the speed of light in vacuum.
- 29.
Numerical value for gold and silver particles vF \(\approx 1.4\;\mbox{ nm}/\mbox{ fs} = 1400\;\mbox{ km}/\mbox{ s}\), Fermi energy EF ≈ 5. 53 eV respectively [22]. In contrast, the typical drift velocity of electrons in an electric wire is of the order of \(\mbox{ mm}/\mbox{ hour}\).
- 30.
Optical damage of metal nanoparticles usually starts at laser energies around 25 GW/cm2 for antenna structures. For single particles this intensity might be doubled because of the lower field enhancement, see e.g. [111].
- 31.
In real-life applications some kind of active control over the properties of the corresponding nanosystem is usually required to achieve signal switching, modulation or amplification, for example. For a passive device these properties are fixed by the nanostructure parameters, in active plasmonics typically hybridized systems (see e.g. [114]) are used, where metallic nanostructures are combined with functional materials.
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Trügler, A. (2016). The World of Plasmons. In: Optical Properties of Metallic Nanoparticles. Springer Series in Materials Science, vol 232. Springer, Cham. https://doi.org/10.1007/978-3-319-25074-8_2
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