So my name is Karl Berggren. I'm gonna talk today about Plasma Oscillations and Plasmons,

and I want to start out by just giving some physical insight into what's going on with

this systems and then I'm gonna do a little bit of back of the envelope algebra to derive

the plasma frequency. Derive might be too strong a term because really, if you want

a proper derivation, you're gonna have to go to another source, you're gonna have to

look into a textbook or another online resource.

But plasmas are very interesting these days and plasma oscillations are interesting because

this particular type of surf ace plasma has recently got a lot of attention in terms of

making nano optical devices and so, that's one of the reasons people are more interested

in this area.

So, what's a plasma first of all? Well, we're mostly gonna talk about metals and metals

as a type of plasma as a gas of charged particles. And a metal and metallic plasma, it's a neutral

plasma so that on the average there's the same number of positive and negative charges

not like I’ve drawn it. In a metal, you have positive charges or screened nuclei that

have the core electrons to still bound them, and then the negative charges are free electrons

that move around. And if you think about what happens in such a neutral system is you can

realize that even on the average it's gonna be neutral maybe over some short-length scales.

So if we just draw a box in here, you may have a little bit extra negative charge and

a little bit extra positive charge, so that there will be occasionally some charge separations.

So let's draw some charge separation happening here. And the charge separation gets at really

what goes on in the plasma, in the plasma oscillation, because once you have a charge

separation you have a Coulomb force.

Now in the plasmon, the positive charges because their nuclei are quite stationary, the electrons

move so in this sytem the force is going to act to pull the negative charge towards the

positive just because electrons are so much lighter, so much more mobile. And as it moves,

it's gonna pick up kinetic energy. In fact, when it passes the positive charge, it's actually

gonna have the maximum kinetic energy and then it's gonna start slowing down because

the Coulomb forces is now gonna be opposite inside and so it will just come out here,

turn around eventually come back and there you get the oscillation. Okay it comes back

and goes through.

Now the oscillation, you can think of oscillations in terms of exchange of energy. So the exchange

of energy here is between an electro-static energy and a kinetic energy. So actually,

the exchange happens twice per oscillation period. So you start out with electro-static

potential, you then transfer the kinetic potential, then back to electro-static potential and

then back to kinetic, and then you complete the cycle. Back to electro-static, so you

go through two exchanges of energy in a single period.

So that's the basic physical picture of charge separation followed by oscillation, and of

course that oscillation is not gonna last forever. There will be a little bit of decay

or loss, and so there will be a little bit of slowing down and it will damp out eventually.

Now, in a more mathematical sense, you can look at the stored potential energy in the

electro-static system. But to do that, first you need to make a guess at the amount of

charge and the separation of the charges. And so, we'll just guess the electron charge

because these are free electrons that are moving. And the separation, the one that's

the most logical is just the average separation of particles in the system we call "S" and

that will be the cube root of the inverse density, or the particle density.

So, if you remember a little bit of your electricity and magnetism, the potential energy stored

between two separated charges is going to be just coulomb’s constant, times the charge

squared, over the separation. And if you remember your harmonic oscillator Physics, so this

is from classical mechanics. And by the way, I'm assuming that you remember electrostatics

and classical mechanics if you studied those. If you haven't, you need to catch up on that

area to understand this. But in that case, the harmonic oscillator energy is one-half

and we call that Omega-P, that's gonna be the oscillation frequency, and X squared,

so that's the standard form for the kinetic, for the energy stored in the oscillator. Not

an X, that should actually be an S, S-squared.

So, seen this as equal as a little bit odd because it's clearly not a truly harmonic

oscillator because the force acting on this goes like one over the separation squared

and it's not proportional to the separation as it should be in a harmonic oscillator,

so that's where this derivation is really quite rough. But it gives you the correct

form, and it gives you the correct physical insight. And the results of a little bit of

algebra is just that the plasma frequency scales with the square-root of the free carrier

density squared, charge squared, and divided by the mass. And this is actually the effect

of mass, not the electron mass. But at this point, we've ignored all sorts of other factors

so like Colomb’s constant has disappeared. And so in fact I'm gonna replace that equality

with just a little proportional to symbol.

So it tells you that the plasma frequency which is typically in the ultra-violet for

metals, it goes up with the free-particle density in the plasma. So that's plasma oscillation.

Now what's a plasmon? Well, plasmon is a single quantum of a plasma oscillation. So just like

a photon is a single quantum of electro-magnetic oscillation, a single plasmon is a quantum

of a plasma oscillation. The difference between electro-magnetic oscillation and plasma oscillation

comes down to this exchange of energy here. So in a conventionalelectro-magnetic oscillation,

you're exchanging energy between electrostatic potential and magnetic potential, magnetically-stored

energy. Whereas in a plasma oscillation, you exchange energy between electrostatic and

kinetic. And of course there's also some magnetic fields formed by the current here. But it's

the presence of this kinetic energy that's really quite different from what you're accustomed

to thinking about in free space for electromagnetic field. And this also relates to the concept

of kinetic inductance which we've talked about in another one of these videos.

So with that, we'll finish for today. If you have any questions, please feel free to leave

them below and I'll do my best to answer them.