## Introduction to bulk and surface plasmon In previous chapter [[Classical Lorentz model]], we mentioned the plasma frequency $\omega_p$, and see that metal have a negative permittivity $\varepsilon'$ between $\omega_{0}$ and $\omega_{p}$. In this chapter we will provide an insight on bulk plasma and surface plasmon, and derived the dispersion relation of bulk plasmon from the lossless Drude model. ### What is plasmon? The word plasmon, ends with "-on", denotes the property of quantum mechanical particles. And indeed, plasmon is a quantum of "electron density wave" (similar concept for spin is [spin density wave](https://en.wikipedia.org/wiki/Spin_density_wave)). It can be imagined as sea of electrons oscillating in metal, and it's a bulk behavior. The natural frequency is the exactly the plasmon frequency (or plasma frequency), $\omega_{p}$, and, $\omega_p^2 = \frac{Ne^2}{\varepsilon_0 m}$ $N$ is electron density, $\varepsilon_{0}$ is vacuum permittivity, $e$ is electron charge, $m$ is mass of electron. ![[Drawing 2024-03-30 23.51.30.excalidraw.svg]] For bulk plasma, the dispersion relation can be estimated based on [[Classical Lorentz model#^b7b0c1|lossless Drude model]] and wave equation. The procedure is provided in the following note. The dispersion relation is $\omega = \sqrt{\omega_{p}^{2}+c^{2}k^{2}}$ This means: - For $\omega = \omega_{p}$, bulk plasmon with $k=0$ (i.e., $\lambda \rightarrow \infty$) - For $\omega < \omega_{p}$, no propagating waves allowed - For $\omega > \omega_{p}$, propagating waves exist Also with $k$ increase, dispersion approaches slope of $c$, which means it's like propagating in vacuum. ($\rightarrow$ the dispersion of light) ![[Drawing 2024-03-31 00.14.45.excalidraw.svg]] >[!Note]- >Consider wave equation $\nabla^2 \mathbf{E} = \mu \varepsilon \frac{\partial^2 \mathbf{E}}{\partial t^2}$, and use plane wave ansatz $\mathbf{E}(\mathbf{r}) = E_0 \exp[i(\mathbf{k} \cdot \mathbf{r} - \omega t)]$, we have the left hand side of the wave equation becomes, >$\nabla^2 \mathbf{E} = -k^2 \mathbf{E} \quad $ >with $\quad k^2 = k_x^2 + k_y^2 + k_z^2$. >And right hand side becomes >$\mu \varepsilon \frac{\partial^2 \mathbf{E}}{\partial t^2} = -\mu \varepsilon \omega^2 \mathbf{E} = -\mu_0 \varepsilon_0 \varepsilon_r(\omega) \omega^2 \mathbf{E}$ >Combining both side, we have >$k^2 \mathbf{E} =\mu_0 \varepsilon_0 \varepsilon_r(\omega) \omega^2 \mathbf{E}$ >$k^2 =\mu_0 \varepsilon_0 \varepsilon_r(\omega) \omega^2 $ >With the approximation $\mu \approx \mu_{0}$ and $\varepsilon_{r}(\omega) = \frac{\varepsilon}{\varepsilon_{0}}$, and recall $c=\frac{1}{\sqrt{\mu_{0}\varepsilon_{0}}}$, we have >$k^2 = \frac{\omega^2}{c^2} \varepsilon_r(\omega)$ >With the lossless Drude model, >$\frac{\varepsilon}{\varepsilon_{0}} = \varepsilon_{r}(\omega) = 1- \frac{\omega_{p}^{2}}{\omega^{2}}$ >we can write >$k^2 = \frac{1}{c^2} (\omega^2 - \omega_p^2)$ > or > $\omega = \sqrt{\omega_p^2 + c^2 k^2}$ ### Surface plasmon Unlike the bulk plasmon, which is the oscillation of all electrons (or say, charge density wave) inside the metal, the surface plasmon is the charge density wave at interface, and it's the quantum of such oscillation. When we say surface plasmon, we often talk about **[[Surface plasmon polaritons|surface plasmon polaritons]]**, or SPPs. The word "polariton" denotes "photon + something". And in the SPPs case, SPP is photon + "surface plasmon", and its a combined excitation. >[!Notice] >Polariton is a kind of **quasiparticle**. Another two types of polaritons are exciton-polaritons (aka **exciton** with photon) and phonon-polaritons (aka **phonon** with photon). >Polaritons belong to Boson. >[!Note] >I'll say the "+" in above expression is more like stating the source of such surface plasmon. >If check the wikipedia, one can see that [The _total_ excitation, including both the charge motion and associated electromagnetic field, is called either a surface plasmon polariton at a planar interface...](https://en.wikipedia.org/wiki/Surface_plasmon#:~:text=The%20total%20excitation%2C%20including%20both%20the%20charge%20motion%20and%20associated%20electromagnetic%20field%2C%20is%20called%20either%20a%20surface%20plasmon%20polariton%20at%20a%20planar%20interface). So SPP is a combined effect. We have the charge motion as SP, as well as the EM field created by this effect. ![[Drawing 2024-03-31 01.03.49.excalidraw.svg]] Surface plasmon polarization propagate along the interface (so it's a guided wave), and its a mixture of photon and electronic motion, so it has different behavior than photon (so different dispersion). And if SPPs are confined in a finite object, like a nano-sized metal sphere, we ill have the surface plasmon localized, and called [[Localized surface plasmons|localized surface plasmons]]. In this case it has a fundamental resonance $\omega_{LSP}$ and has specific resonance for give size. Surface plasmon are not limited by diffractions like photons, and it can be concentrated at interface or focus at a small volume (nano-sized metal for LSPs), this provides huge aspects of approximations in sensor, catalyst, etc. But metal are intrinsically optically lossy, and electron motion generates heat (although this could provide some application), and loss increases with the confinement, for LSPs. >[!Info] >See more at >[[Surface plasmon polaritons]] and [[Localized surface plasmons]], as well as wiki pages: >https://en.wikipedia.org/wiki/Surface_plasmon >https://en.wikipedia.org/wiki/Localized_surface_plasmon >https://en.wikipedia.org/wiki/Surface_plasmon_polariton