Plasma oscillation

Plasma oscillations, also known as Langmuir waves (after Irving Langmuir), are rapid oscillations of the electron density in conducting media such as plasmas or metals in the ultraviolet region. The oscillations can be described as an instability in the dielectric function of a free electron gas. The frequency depends only weakly on the wavelength of the oscillation. The quasiparticle resulting from the quantization of these oscillations is the plasmon.

Langmuir waves were discovered by American physicists Irving Langmuir and Lewi Tonks in the 1920s.^[1] They are parallel in form to Jeans instability waves, which are caused by gravitational instabilities in a static medium.

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Plasmas Oscillations and plasmons explained 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.

Mechanism

Consider an electrically neutral plasma in equilibrium, consisting of a gas of positively charged ions and negatively charged electrons. If one displaces by a tiny amount an electron or a group of electrons with respect to the ions, the Coulomb force pulls the electrons back, acting as a restoring force.

'Cold' electrons

If the thermal motion of the electrons is ignored, it is possible to show that the charge density oscillates at the plasma frequency

\omega _{\mathrm {pe} }={\sqrt {\frac {n_{\mathrm {e} }e^{2}}{m^{*}\varepsilon _{0}}}},\left[\mathrm {rad/s} \right]

(SI units),

\omega _{\mathrm {pe} }={\sqrt {\frac {4\pi n_{\mathrm {e} }e^{2}}{m^{*}}}},\left[\mathrm {rad/s} \right]

(cgs units),

where $n_{\mathrm {e} }$ is the number density of electrons, $e$ is the electric charge, $m^{*}$ is the effective mass of the electron, and $\varepsilon _{0}$ is the permittivity of free space. Note that the above formula is derived under the approximation that the ion mass is infinite. This is generally a good approximation, as the electrons are so much lighter than ions.

Proof using Maxwell equations.^[2] Assuming charge density oscillations $\rho (\omega )=\rho _{0}e^{-i\omega t}$ the continuity equation:

\nabla \cdot \mathbf {j} =-{\frac {\partial \rho }{\partial t}}=i\omega \rho (\omega )

the Gauss law

\nabla \cdot \mathbf {E} (\omega )=4\pi \rho (\omega )

and the conductivity

\mathbf {j} (\omega )=\sigma (\omega )\mathbf {E} (\omega )

taking the divergence on both sides and substituting the above relations:

i\omega \rho (\omega )=4\pi \sigma (\omega )\rho (\omega )

which is always true only if

1+{\frac {4\pi i\sigma (\omega )}{\omega }}=0

But this is also the dielectric constant (see Drude Model)

\epsilon (\omega )=1+{\frac {4\pi i\sigma (\omega )}{\omega }}

and the condition of transparency (i.e.

\epsilon \geq 0

from a certain plasma frequency

\omega _{\rm {p}}

and above), the same condition here

\epsilon =0

apply to make possible also the propagation of density waves in the charge density.

This expression must be modified in the case of electron-positron plasmas, often encountered in astrophysics.^[3] Since the frequency is independent of the wavelength, these oscillations have an infinite phase velocity and zero group velocity.

Note that, when $m^{*}=m_{\mathrm {e} }$ , the plasma frequency, $\omega _{\mathrm {pe} }$ , depends only on physical constants and electron density $n_{\mathrm {e} }$ . The numeric expression for angular plasma frequency is

f_{\text{pe}}={\frac {\omega _{\text{pe}}}{2\pi }}~\left[{\text{Hz}}\right]

Metals are only transparent to light with a frequency higher than the metal's plasma frequency. For typical metals such as aluminium or silver, $n_{\mathrm {e} }$ is approximately 10²³ cm⁻³, which brings the plasma frequency into the ultraviolet region. This is why most metals reflect visible light and appear shiny.

'Warm' electrons

When the effects of the electron thermal speed ${\textstyle v_{\mathrm {e,th} }={\sqrt {k_{\mathrm {B} }T_{\mathrm {e} }/m_{\mathrm {e} }}}}$ are taken into account, the electron pressure acts as a restoring force as well as the electric field and the oscillations propagate with frequency and wavenumber related by the longitudinal Langmuir^[4] wave:

\omega ^{2}=\omega _{\mathrm {pe} }^{2}+{\frac {3k_{\mathrm {B} }T_{\mathrm {e} }}{m_{\mathrm {e} }}}k^{2}=\omega _{\mathrm {pe} }^{2}+3k^{2}v_{\mathrm {e,th} }^{2},

called the Bohm–Gross dispersion relation. If the spatial scale is large compared to the Debye length, the oscillations are only weakly modified by the pressure term, but at small scales the pressure term dominates and the waves become dispersionless with a speed of

{\sqrt {3}}\cdot v_{\mathrm {e,th} }

. For such waves, however, the electron thermal speed is comparable to the phase velocity, i.e.,

v\sim v_{\mathrm {ph} }\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\omega }{k}},

so the plasma waves can accelerate electrons that are moving with speed nearly equal to the phase velocity of the wave. This process often leads to a form of collisionless damping, called Landau damping. Consequently, the large-k portion in the dispersion relation is difficult to observe and seldom of consequence.

In a bounded plasma, fringing electric fields can result in propagation of plasma oscillations, even when the electrons are cold.

In a metal or semiconductor, the effect of the ions' periodic potential must be taken into account. This is usually done by using the electrons' effective mass in place of m.

Plasma oscillations and the effect of the negative mass

Plasma oscillations may give rise to the effect of the “negative mass”. The mechanical model giving rise to the negative effective mass effect is depicted in Figure 1. A core with mass $m_{2}$ is connected internally through the spring with constant $k_{2}$ to a shell with mass $m_{1}$ . The system is subjected to the external sinusoidal force $F(t)={\widehat {F}}\sin \omega t$ . If we solve the equations of motion for the masses $m_{1}$ and $m_{2}$ and replace the entire system with a single effective mass $m_{\rm {eff}}$ we obtain:^[5]^[6]^[7]^[8]^[9]

m_{\rm {eff}}=m_{1}+{m_{2}\omega _{0}^{2} \over \omega _{0}^{2}-\omega ^{2}},

where

{\textstyle \omega _{0}={\sqrt {k_{2}/m_{2}}}}

. When the frequency

\omega

approaches

\omega _{0}

from above the effective mass

m_{\rm {eff}}

will be negative.^[5]^[6]^[7]^[8]

The negative effective mass (density) becomes also possible based on the electro-mechanical coupling exploiting plasma oscillations of a free electron gas (see Figure 2).^[9]^[10] The negative mass appears as a result of vibration of a metallic particle with a frequency of $\omega$ which is close the frequency of the plasma oscillations of the electron gas $m_{2}$ relatively to the ionic lattice $m_{1}$ . The plasma oscillations are represented with the elastic spring $k_{2}=\omega _{\rm {p}}^{2}m_{2}$ , where $\omega _{\rm {p}}$ is the plasma frequency. Thus, the metallic particle vibrated with the external frequency ω is described by the effective mass

m_{\rm {eff}}=m_{1}+{m_{2}\omega _{\rm {p}}^{2} \over \omega _{\rm {p}}^{2}-\omega ^{2}},

which is negative when the frequency

\omega

approaches

\omega _{\rm {p}}

from above. Metamaterials exploiting the effect of the negative mass in the vicinity of the plasma frequency were reported.^[9]^[10]

References

^ Tonks, Lewi; Langmuir, Irving (1929). "Oscillations in ionized gases" (PDF). Physical Review. 33 (8): 195–210. Bibcode:1929PhRv...33..195T. doi:10.1103/PhysRev.33.195. PMC 1085653.
^ Ashcroft, Neil; Mermin, N. David (1976). Solid State Physics. New York: Holt, Rinehart and Winston. p. 19. ISBN 978-0-03-083993-1.
^ Fu, Ying (2011). Optical properties of nanostructures. Pan Stanford. p. 201.
^ *Andreev, A. A. (2000), An Introduction to Hot Laser Plasma Physics, Huntington, New York: Nova Science Publishers, Inc., ISBN 978-1-56072-803-0
^ ^a ^b Milton, Graeme W; Willis, John R (2007-03-08). "On modifications of Newton's second law and linear continuum elastodynamics". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 463 (2079): 855–880. Bibcode:2007RSPSA.463..855M. doi:10.1098/rspa.2006.1795. S2CID 122990527.
^ ^a ^b Chan, C. T.; Li, Jensen; Fung, K. H. (2006-01-01). "On extending the concept of double negativity to acoustic waves". Journal of Zhejiang University Science A. 7 (1): 24–28. doi:10.1631/jzus.2006.A0024. ISSN 1862-1775. S2CID 120899746.
^ ^a ^b Huang, H. H.; Sun, C. T.; Huang, G. L. (2009-04-01). "On the negative effective mass density in acoustic metamaterials". International Journal of Engineering Science. 47 (4): 610–617. doi:10.1016/j.ijengsci.2008.12.007. ISSN 0020-7225.
^ ^a ^b Yao, Shanshan; Zhou, Xiaoming; Hu, Gengkai (2008-04-14). "Experimental study on negative effective mass in a 1D mass–spring system". New Journal of Physics. 10 (4): 043020. Bibcode:2008NJPh...10d3020Y. doi:10.1088/1367-2630/10/4/043020. ISSN 1367-2630.
^ ^a ^b ^c Bormashenko, Edward; Legchenkova, Irina (April 2020). "Negative Effective Mass in Plasmonic Systems". Materials. 13 (8): 1890. Bibcode:2020Mate...13.1890B. doi:10.3390/ma13081890. PMC 7215794. PMID 32316640.

Text was copied from this source, which is available under a Creative Commons Attribution 4.0 International License.
^ ^a ^b Bormashenko, Edward; Legchenkova, Irina; Frenkel, Mark (August 2020). "Negative Effective Mass in Plasmonic Systems II: Elucidating the Optical and Acoustical Branches of Vibrations and the Possibility of Anti-Resonance Propagation". Materials. 13 (16): 3512. Bibcode:2020Mate...13.3512B. doi:10.3390/ma13163512. PMC 7476018. PMID 32784869.

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