Abstract
The book aims to present a systemic and self-contained guide to the canonical electromagnetic and electrostatic boundary-value problems in metallic nanostructures. In this way, the conduction electrons of a metallic medium are modeled as a degenerate electron gas, whose dynamics may be described by means of the hydrodynamic theory. Therefore, at first we need to know something about the hydrodynamic model of an electron gas. Then, we need to know something about the basic concepts and formalism of electromagnetic and electrostatic theories of an electron gas that will be used later in the book. For brevity, in many sections of this chapter the \(\exp (-i\omega t)\) time factor is suppressed. Furthermore, all media under consideration are nonmagnetic and attention is only confined to the linear phenomena.
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12 November 2020
This book was inadvertently published without updating the following corrections.
Notes
- 1.
We use the term metallic structures throughout this book, while keeping in mind the analysis can be applied to highly doped semiconductors and potentially other conducting systems.
- 2.
In solid state physics the quanta of longitudinal charge density waves is called a plasmon . The first theoretical description of plasmons was presented by Pines and Bohm in 1952 [5]. In plasma physics these charge density oscillations are referred to as plasma waves.
- 3.
This model has not been applied yet to practical MNSs.
- 4.
e ≈ 1.60 × 10−19 C, and m e ≈ 9.11 × 10−31 kg.
- 5.
ε 0 ≈ 8.854 × 10−12 F/m.
- 6.
h ≈ 6.63 × 10−34 J.s ≈ 4.14 × 10−15 eV.s.
- 7.
The equation of drift motion is also known as the hydrodynamic equation or Newton’s equation of motion.
- 8.
In general, we have \(\gamma \rightarrow \gamma +A\dfrac {v_{F}}{R}\), where A is a constant, which is related to the probability of the free electrons scattering off the surface of the MNS. However, experimental observations and advanced theoretical calculations show that in most cases A ≈ 1.
- 9.
Thus, the present SHD model yields a coefficient that is by a factor 9∕5 too small.
- 10.
We note that for typical metals the values of α are of the order of the Bohr velocity.
- 11.
Materials whose constitutive parameters (for instant dielectric function and conductivity, which are, in general, functions of the applied field strength, the position within the medium, the direction of the applied field and the frequency of operation) are not functions of the applied field are usually known as linear; otherwise they are nonlinear.
- 12.
When the constitutive parameters of media are not functions of position, the materials are called homogeneous.
- 13.
μ 0 = 4π × 10−7 H/m.
- 14.
When the medium is isotropic, the vector J and E are colinear.
- 15.
The Fourier transform can be defined in different ways. We define the Fourier transforms in 3D with respect to position and time and their inverses in the following way:
$$\displaystyle \begin{aligned}f(\mathbf{k})=\int f(\mathbf{r})e^{-i\mathbf{k}\cdot \mathbf{r}}\mathop{}\!\mathrm{d}\mathbf{r}\;, \;\;\;\;\;\;\; f(\mathbf{r})=\dfrac{1}{(2\pi)^{3}}\int f(\mathbf{k}) e^{i\mathbf{k}\cdot \mathbf{r}}\mathop{}\!\mathrm{d}\mathbf{k}\;,\end{aligned}$$$$\displaystyle \begin{aligned}f(\omega)=\int f(t)e^{i\omega t}\mathop{}\!\mathrm{d} t\;, \;\;\;\;\;\;\; f(t)=\dfrac{1}{2\pi}\int f(\omega)e^{-i\omega t}\mathop{}\!\mathrm{d}\omega\;. \end{aligned}$$ - 16.
We indicate a tensor by bold italics but, where Greek characters are used, as in the above case, we add underlining for additional clarity.
- 17.
- 18.
Here, this is an electron plasma that is not subjected to a steady imposed magnetic field.
- 19.
We note that the present study is based on the non-relativistic QHD model, where the phase velocities of the waves (as well as the particle velocities) are non-relativistic. In relativistic EGs, the relativistic effects may greatly modify the behavior of plasma waves.
- 20.
However, in an anisotropic EG we can still have a steady imposed magnetic field B 0.
- 21.
In the literature, the values of plasma frequency are usually given by ν p = ω p∕2π.
- 22.
When the index of refraction n goes to zero, we say that there is a cutoff. This occurs when the phase velocity v p = ω∕k goes to infinity.
- 23.
When the index of refraction n goes to infinity, we say that there is a resonance. This occurs when the phase velocity goes to zero.
- 24.
The tangential components of these field vectors to the boundary surface are still vectors in the tangential plane of the surface.
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Moradi, A. (2020). Basic Concepts and Formalism. In: Canonical Problems in the Theory of Plasmonics. Springer Series in Optical Sciences, vol 230. Springer, Cham. https://doi.org/10.1007/978-3-030-43836-4_1
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