Plasmon and plasma oscillation

From Wikipedia, the free encyclopedia

In physics, a plasmon is a quantum of plasma oscillation (see below). Just as light (an optical oscillation) consists of photons, the plasma oscillation consists of plasmons. The plasmon can be considered as a quasiparticle since it arises from the quantization of plasma oscillations, just like phonons are quantizations of mechanical vibrations. Thus, plasmons are collective (a discrete number) oscillations of the free electron gas density. For example, at optical frequencies, plasmons can couple with a photon to create another quasiparticle called a plasmon polariton.

Derivation

The plasmon was initially proposed in 1952 by David Pines and David Bohm and was shown to arise from a Hamiltonian for the long-range electron-electron correlations.

Since plasmons are the quantization of classical plasma oscillations, most of their properties can be derived directly from Maxwell's equations.

Explanation

Plasmons can be described in the classical picture as an oscillation of electron density with respect to the fixed positive ions in a metal. To visualize a plasma oscillation, imagine a cube of metal placed in an external electric field pointing to the right. Electrons will move to the left side (uncovering positive ions on the right side) until they cancel the field inside the metal. If the electric field is removed, the electrons move to the right, repelled by each other and attracted to the positive ions left bare on the right side. They oscillate back and forth at the plasma frequency until the energy is lost in some kind of resistance or damping. Plasmons are a quantization of this kind of oscillation.

Role

Plasmons play a large role in the optical properties of metals and semiconductors. Light of frequencies below the plasma frequency is reflected by a material because the electrons in the material screen the electric field of the light. Light of frequencies above the plasma frequency is transmitted by a material because the electrons in the material cannot respond fast enough to screen it. In most metals, the plasma frequency is in the ultraviolet, making them shiny (reflective) in the visible range. Some metals, such as copper and gold, have electronic inter-band transitions in the visible range, whereby specific light energies (colors) are absorbed, yielding their distinct color. In semiconductors, the valence electron plasmon frequency is usually in the deep ultraviolet, while their electronic inter-band transitions are in the visible range, whereby specific light energies (colors) are absorbed, yielding their distinct color which is why they are reflective. It has been shown that the plasmon frequency may occur in the mid-infrared and near-infrared region when semiconductors are in the form of nanoparticles with heavy doping.

The plasmon energy can often be estimated in the free electron model as

${\displaystyle E_{p}=}$${\displaystyle \hbar }$${\displaystyle {\sqrt {\frac {ne^{2}}{m\epsilon _{0}}}}=}$${\displaystyle \hbar }$${\displaystyle \omega _{p},}$

where ${\displaystyle n}$ is the conduction electron density, ${\displaystyle e}$ is the elementary charge, ${\displaystyle m}$ is the electron mass, ${\displaystyle \epsilon _{0}}$ the permittivity of free space, ${\displaystyle \hbar }$ the reduced Planck constant and ${\displaystyle \omega _{p}}$ the plasmon frequency.

Surface plasmons

Surface plasmons are those plasmons that are confined to surfaces and that interact strongly with light resulting in a polariton. They occur at the interface of a material exhibiting positive real part of their relative permittivity, i.e. dielectric constant, (e.g. vacuum, air, glass and other dielectrics) and a material whose real part of permittivity is negative at the given frequency of light, typically a metal or heavily doped semiconductors. In addition to opposite sign of the real part of the permittivity, the magnitude of the real part of the permittivity in the negative permittivity region should typically be larger than the magnitude of the permittivity in the positive permittivity region, otherwise the light is not bound to the surface (i.e. the surface plasmons do not exist) as shown in the famous book by Raether. At visible wavelengths of light, e.g. 632.8 nm wavelength provided by a He-Ne laser, interfaces supporting surface plasmons are often formed by metals like silver or gold (negative real part permittivity) in contact with dielectrics such as air or silicon dioxide. The particular choice of materials can have a drastic effect on the degree of light confinement and propagation distance due to losses. Surface plasmons can also exist on interfaces other than flat surfaces, such as particles, or rectangular strips, v-grooves, cylinders, and other structures. Many structures have been investigated due to the capability of surface plasmons to confine light below the diffraction limit of light. 

Surface plasmons can play a role in surface-enhanced Raman spectroscopy and in explaining anomalies in diffraction from metal gratings (Wood's anomaly), among other things. Surface plasmon resonance is used by biochemists to study the mechanisms and kinetics of ligands binding to receptors (i.e. a substrate binding to an enzyme). Multi-parametric surface plasmon resonance can be used not only to measure molecular interactions, but also nanolayer properties or structural changes in the adsorbed molecules, polymer layers or graphene, for instance. 

Surface plasmons may also be observed in the X-ray emission spectra of metals. A dispersion relation for surface plasmons in the X-ray emission spectra of metals has been derived (Harsh and Agarwal).

Gothic stained glass rose window of Notre-Dame de Paris. The colors were achieved by colloids of gold nano-particles.

 

More recently surface plasmons have been used to control colors of materials. This is possible since controlling the particle's shape and size determines the types of surface plasmons that can couple to it and propagate across it. This in turn controls the interaction of light with the surface. These effects are illustrated by the historic stained glass which adorn medieval cathedrals. In this case, the color is given by metal nanoparticles of a fixed size which interact with the optical field to give the glass its vibrant color. In modern science, these effects have been engineered for both visible light and microwave radiation. Much research goes on first in the microwave range because at this wavelength material surfaces can be produced mechanically as the patterns tend to be of the order a few centimeters. To produce optical range surface plasmon effects involves producing surfaces which have features less than 400 nm. This is much more difficult and has only recently become possible to do in any reliable or available way. 

Recently, graphene has also been shown to accommodate surface plasmons, observed via near field infrared optical microscopy techniques and infrared spectroscopy. Potential applications of graphene plasmonics mainly addressed the terahertz to mid-infrared frequencies, such as optical modulators, photodetectors, biosensors.

Possible applications

The position and intensity of plasmon absorption and emission peaks are affected by molecular adsorption, which can be used in molecular sensors. For example, a fully operational device detecting casein in milk has been prototyped, based on detecting a change in absorption of a gold layer. Localized surface plasmons of metal nanoparticles can be used for sensing different types of molecules, proteins, etc. 

Plasmons are being considered as a means of transmitting information on computer chips, since plasmons can support much higher frequencies (into the 100 THz range, whereas conventional wires become very lossy in the tens of GHz). However, for plasmon-based electronics to be practical, a plasmon-based amplifier analogous to the transistor, called a plasmonstor, needs to be created.

Plasmons have also been proposed as a means of high-resolution lithography and microscopy due to their extremely small wavelengths; both of these applications have seen successful demonstrations in the lab environment. 

Finally, surface plasmons have the unique capacity to confine light to very small dimensions, which could enable many new applications. 

Surface plasmons are very sensitive to the properties of the materials on which they propagate. This has led to their use to measure the thickness of monolayers on colloid films, such as screening and quantifying protein binding events. Companies such as Biacore have commercialized instruments that operate on these principles. Optical surface plasmons are being investigated with a view to improve makeup by L'Oréal and others.

In 2009, a Korean research team found a way to greatly improve organic light-emitting diode efficiency with the use of plasmons.

A group of European researchers led by IMEC has begun work to improve solar cell efficiencies and costs through incorporation of metallic nanostructures (using plasmonic effects) that can enhance absorption of light into different types of solar cells: crystalline silicon (c-Si), high-performance III-V, organic, and dye-sensitized.  However, for plasmonic photovoltaic devices to function optimally, ultra-thin transparent conducting oxides are necessary. Full color holograms using plasmonics have been demonstrated. 

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Plasma oscillation

From Wikipedia, the free encyclopedia

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 only depends 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. They are parallel in form to Jeans instability waves, which are caused by gravitational instabilities in a static medium.

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

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

${\displaystyle \omega _{\mathrm {pe} }={\sqrt {\frac {4\pi n_{\mathrm {e} }e^{2}}{m^{*}}}},[rad/s]}$ (cgs units),

where ${\displaystyle n_{\mathrm {e} }}$ is the number density of electrons, e is the electric charge, m^* is the effective mass of the electron, and ${\displaystyle \varepsilon _{0}}$ 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. (This expression must be modified in the case of electron-positron plasmas, often encountered in astrophysics). Since the frequency is independent of the wavelength, these oscillations have an infinite phase velocity and zero group velocity.

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

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

Metals are only transparent to light with frequency higher than the metal's plasma frequency. For typical metals such as copper or silver, ${\displaystyle n_{\mathrm {e} }}$ 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 ${\displaystyle v_{\mathrm {e,th} }={\sqrt {\frac {k_{\mathrm {B} }T_{\mathrm {e} }}{m_{\mathrm {e} }}}}}$ 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 wave:

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

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 ${\displaystyle {\sqrt {3}}\cdot v_{\mathrm {e,th} }}$. For such waves, however, the electron thermal speed is comparable to the phase velocity, i.e.,

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

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.