mexicankrill

Scientist. Shitposter. Sleepy.

Precursor to Plasmons

In this post, I will introduce the physics to understand plasmons, which will act as a starting point for sharing my research. Posts on this topic will be written chronologically, such that necessary knowledge is entirely contained in earlier posts. Some basic prior knowledge is expected, but I will make it explicitly known at the start of each post.

Whilst my research is conducted on nanoscale structures, it is possible to understand the behavior of plasmons without needing to consider quantum effects. I will instead use electrostatic analysis of charge density fluctuations in an electron gas based on David Pines and David Bohm's work in the early 50s [1, 2]. To follow this, I assume that you understand basic electrostatics and Fourier analysis.

We will consider the electron gas to behave as an aggregate of free electrons within a medium of fixed positive charges with equal mean density to the electrons, such that the overall charge of the system is zero. This serves as the basis for our model. Each electron experiences a force equal to the sum of forces due to the other electrons, as well as from the positive charges. We can then write the Fourier series of the potential energy between two electrons labelled \(i\) and \(j\) as \begin{equation} \frac{e^{2}}{\lvert{\mathbf{x}}_{i}-\mathbf{x}_{j}\rvert}=4\pi{e}^{2}\sum_{k}\frac{1}{k^{2}}e^{i\mathbf{k}\cdot(\mathbf{x}_{i}-\mathbf{x}_{j})} \end{equation} The equation of motion of electron \(i\) is then \begin{equation} \ddot{\mathbf{x}}_{i}=-\frac{4\pi{e}^{2}i}{m}\sum_{j,k}\frac{\mathbf{k}}{k^{2}}e^{i\mathbf{k}\cdot(\mathbf{x}_{i}-\mathbf{x}_{j})} \label{eq:motion-1} \end{equation} This is near impossible to solve by consideration of the motion of individual particles, especially in dense electron gases where collisions between multiple bodies cannot be neglected. Furthermore, the Coulomb potential has significant enough range that such collisions must be considered in lower density electron gases. In these conditions, the electrons are known to behave as a collective, producing plasma oscillations. This allows us to attack the equations of motion using the organised movement of the plasma as a starting point to reach an accurate solution.

We will assume that our electrons can be treated as point particles, such that the particle density is \begin{equation} \rho(\mathbf{x})=\sum_{i}\delta(\mathbf{x}-\mathbf{x}_i) \end{equation} where \(\delta(x)\) is the Dirac delta function. Instead of directly working with the particle density, we use its Fourier components, defined as \begin{equation} \rho_{k}=\int{d\mathbf{x}}\rho(\mathbf{x})e^{-\mathbf{k}\cdot\mathbf{x}}=\sum_{i}e^{-\mathbf{k}\cdot\mathbf{x}_{i}} \label{eq:density-fourier} \end{equation} with \begin{equation} \rho(\mathbf{x})=\sum_{i,k}e^{-\mathbf{k}\cdot\mathbf{x}_{i}}. \end{equation} Trivially, \(\rho_{0} = n\) is the mean electron density, with higher order \(\rho_{k}\) terms representing density fluctuations about \(\rho_{0}\). From this result, the equation of motion eqn \ref{eq:motion-1} becomes \begin{equation} \ddot{\mathbf{x}}_{i}=-\frac{4\pi{e}^{2}i}{m}\sum_{k}\frac{\mathbf{k}}{k^{2}}\rho_{k}e^{i\mathbf{k}\cdot\mathbf{x}_{i}} \end{equation} hence the \(\rho_{k}\) determine the force which acts on each particle.

Now we will consider the time dependent behavior of \(\rho_{k}\) by differentiation of eqn \ref{eq:density-fourier}, giving \begin{equation} \dot{\rho}_{k}=-i\sum_{i}(\mathbf{k}\cdot\mathbf{v}_{i})e^{-i\mathbf{k}\cdot\mathbf{x}_{i}} \end{equation} and \begin{equation} \partial^{2}_{t}\rho_{k}=-\sum_{i}\left[(\mathbf{k}\cdot\mathbf{v}_{i})^{2}+i\mathbf{k}\cdot\dot{\mathbf{v}}_{i}\right]e^{-i\mathbf{k}\cdot\mathbf{x}_{i}}. \end{equation} We get \(\dot{\mathbf{v}}_{i}\) from the equation of motion eqn \ref{eq:motion-1}, so \(\partial^{2}_{t}\rho_{k}\) becomes \begin{equation} \partial^{2}_{t}\rho_{k}=-\sum_{i}(\mathbf{k}\cdot\mathbf{v}_{i})^{2}e^{-i\mathbf{k}\cdot\mathbf{x}_{i}} -\sum_{k',i,j|k'\neq{0}}\frac{4\pi{e}^{2}}{m(k')^{2}}\mathbf{k}\cdot\mathbf{k}' e^{i(\mathbf{k}'-\mathbf{k})\cdot\mathbf{x}_{i}}e^{-i\mathbf{k}'\cdot\mathbf{x}_{j}}. \end{equation}

The second term complicates the analysis quite a bit, however we can make use of some careful approximations to simplify. We can split the sum over \(\mathbf{k}'\) firstly over the terms with \(\mathbf{k}'=\mathbf{k}\), which is independent of the particle coordinate \(\mathbf{x}_{i}\), so that summing over the index \(i\) yields the total number of particles, \(n\). Then considering the terms with \(\mathbf{k}'\neq\mathbf{k}\), which have a phase-induced factor, \(e^{i(\mathbf{k}'-\mathbf{k})\cdot\mathbf{x}_{i}}\), which has position dependence. Now we introduce the random phase approximation, which allows us to assume that these phase factors average out to zero, since we have an approximately random distribution of particle positions. This allows us to neglect these terms, hence \begin{equation} \partial^{2}_{t}\rho_{k}=-\sum_{i}(\mathbf{k}\cdot\mathbf{v}_{i})^{2}e^{-i\mathbf{k}\cdot\mathbf{x}_{i}} -\frac{4\pi{n}e^{2}}{m}\sum_{i}e^{-i\mathbf{k}\cdot\mathbf{x}_{i}} \label{eq:d2pdt2} \end{equation}

The first term is independent of particle interactions, and is simply due to random thermal motion of individual particles. The second term describes the effects of particle interactions. If we take \(\mathbf{k}\) to be sufficiently small, the effects of thermal fluctuation are negligible against the particle interactions. From this assumption, eqn \ref{eq:d2pdt2} becomes \begin{equation} \partial^{2}_{t}\rho_{k}+\frac{4\pi{n}e^{2}}{m}\rho_{k}=0. \end{equation} Hence, the Coulomb interaction between electrons gives rise to harmonic oscillation of the plasma with frequency \begin{equation} \omega_{P}=\left(\frac{4\pi{n}e^{2}}{m}\right)^{1/2}. \end{equation} We see that excitation of a given Fourier component \(\rho_{k}\) corresponds to a wave-like harmonic oscillation of the plasma density, similar to a phonon, so we call this a plasmon!

Whilst this would make an appropriate place to stop, I would like us to also consider what is the physical significance of each term contributing to the temporal variation of \(\rho_{k}\). First we may imagine the same system in the absence of any Coulomb interactions, then the particles will move in a straight line with constant velocity \(\mathbf{v}_{0,i}\) and the position is \(\mathbf{x}_{i}=\mathbf{x}_{0,i}+\mathbf{v}_{0,i}t\). Then the Fourier component is \begin{equation} \rho_{0,k}=\sum_{i}e^{i\mathbf{k}\cdot(\mathbf{x}_{0,i}+\mathbf{v}_{0,i}t)}. \end{equation} For sufficiently large \(\rho_{0,k}\), it is necessary that the positions \(\mathbf{x}_{0,i}\) are distributed such that the terms \(e^{i\mathbf{k}\cdot\mathbf{x}_{0,i}}\) are generally in phase. However, these terms will oscillate with frequency \(\mathbf{k}\cdot\mathbf{v}_{0,i}\), which is likely to be different for each particle. Hence, even if all the particles at some point have the same phase, they will soon fall out of phase, and their sum will again be zero as a consequence of the random phase approximation. Thus, in a gas of free particles, we cannot expect a appreciable density fluctuation for any significant length of time. This leads us to conclude that a collection of free particles cannot show organised behavior, and that any disturbance will quickly disappear due to random diffusion.

On the other hand, the Coulomb interaction in eqn \ref{eq:d2pdt2} causes the contribution from each particle on \(\partial^{2}_{t}\rho_{k}\) to oscillate at the same frequency as any other particle. So, in the absence of random thermal motion, we would observe perfectly organised oscillations of \(\rho_{k}\). Naturally this is not the case, and both contributions are simultaneously present, so the net behavior will show some aspects of the collective organised motion, as well as some aspects of a gas of randomly moving free particles. Thus, for our collective description to be accurate, we require that the Coulomb interactions are dominant over the random thermal motion, \begin{equation} \frac{4\pi{n}e^{2}}{m}\gg\langle(\mathbf{k}\cdot\mathbf{v}_{i})^{2}\rangle. \end{equation} From this we see that organised oscillation is greatly favored by a high particle density. The \(e^{2}\) factor measures the Coulomb force interaction strength, so a stronger interaction also favors organised behavior. We expect the velocities of an electron gas to follow a Maxwell distribution, so this criterion becomes \begin{equation} k^{2}\ll\frac{12\pi{n}e^{2}}{m\langle\mathbf{v}_{i}^{2}\rangle}=\frac{4\pi{n}e^{2}}{k_{b}T}=\lambda_{D}^{-2} \end{equation} where \(k_{B}\) is Boltzmann's constant, and \(\lambda_{D}\) is the Debye length. Hence \begin{equation} \lambda_{D}=\frac{k_{B}T}{4\pi{n}e^{2}}=\frac{1}{3}\frac{\langle\mathbf{v}_{i}^{2}\rangle}{\omega_{P}^{2}}. \end{equation} This gives us an idea as to what length scales better support which model. At distances greater than \(\lambda_{D}\), organised behavior is most important, whilst shorter distances favor consideration of individual particles.

Whilst this is hardly scratches the surface about plasmons, I believe what is contained within this post serves as a good introduction to them and why they should exist in the first place. Pines and Bohm go into much deeper analysis in their 1952 paper mentioned earlier [1], so I highly encourage anybody interested to read those for more information, though be warned it contains notably harder mathematics than we have used here. In addition to the extra analysis, they also justify why we may use the random phase approximation. For the very adventurous, there is also a followup to this paper from 1953 [2] which uses quantum mechanical arguments to describe the collective behavior. Whilst interesting, I do not believe it is very enlightening for the average reader, even if you are able to understand the mathematics.

With that I will conclude this post. Congratulations if you've made it to the end, and thank you for reading. If you would like to learn more, this is only the first post in a series about plasmonics, where I eventually build up to discussion about my research, so I encourage you to read Part II.

Max


References

[1] D. Pines, D. Bohm. "A Collective Description of Electron Interactions: II. Collective vs Individual Particle Aspects of the Interactions," in Phys. Rev., vol. 85, pp. 338–353, 1952.

[2] D. Bohm, D. Pines. "A Collective Description of Electron Interactions: III. Coulomb Interactions in a Degenerate Electron Gas," in Phys. Rev., vol. 92, pp. 609–625, 1953.


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