# Surface plasmon

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Surface plasmon

Surface plasmons, also referred to in the literature as surface plasma polaritons, are fluctuations in the electron density at the boundary of two materials. Plasmons are the collective vibrations of an electron gas (or plasma) surrounding the atomic lattice sites of a metal. When plasmons couple with a photon, the resulting particle is called a polariton. This polariton propagates along the surface of the metal until it decays, either by absorption, whereupon the energy is converted into phonons, or by a radiative transition into a photon. Surface Plasmons were first reported in 1957 by R.H. Ritchie [cite journal |last=Ritchie |first=R. H. |month=June |year=1957 |title= Plasma Losses by Fast Electrons in Thin Films |journal=Physical Review |volume=106 |issue=5 |pages=874–881 |doi=10.1103/PhysRev.106.874] . In the following two decades, Surface Plasmons were extensively studied by many scientists, the foremost of whom were Heinz Raether, E. Kretschmann, and A. Otto.

Excitation

Surface plasmons can be excited by both electrons and photons. Excitation by electrons is created by shooting electrons into a metal. As the electrons scatter, energy is transferred into the plasma. The component of the scattering vector parallel to the surface results in the formation of a surface plasmon.

Excitation by photons requires the use of a coupling medium such as a prism or grating to match the photon and surface plasmon wave vectors. A prism can be positioned against a thin metal film in the Kretschmann configuration or very close to a metal surface in the Otto configuration (Figure 1). A grating coupler matches the wave vectors by increasing the wave vector by the spatial frequency of the grating (Figure 2). This method, while less frequently utilized, is critical to the theoretical understanding of the impact of surface roughness.

Dispersion relationship

The electric field of a propagating electromagnetic wave can be expressed

: $E= E_\left\{0\right\}exp \left[i\left(k_\left\{x\right\} x + k_\left\{z\right\} z -omega t\right)\right] ,$

where "k" is the wave number and &omega; is the frequency of the wave. By solving Maxwell's equations for the electromagnetic wave at an interface between two materials with relative dielectric constants $varepsilon_1$ and $varepsilon_2$ (see figure 3) with the appropriate continuity relationships the boundary conditions are found to becite book |last=Raether |first=Heinz |year=1988 |title=Surface Plasmons on Smooth and Rough Surfaces and on Gratings |location=New York |publisher=Springer-Verlag |isbn=0-387-17363-3| series=Springer Tracts in Modern Physics 111 (Germany: ISBN 3-540-17363-3)]

: $frac\left\{k_\left\{z1\left\{varepsilon_1\right\} + frac\left\{k_\left\{z2\left\{varepsilon_2\right\} = 0$

and

: $k_\left\{x\right\}^2+k_\left\{zi\right\}^2=varepsilon_i left\left(frac\left\{omega\right\}\left\{c\right\} ight\right)^2 qquad i=1,2$

where "c" is the speed of light in a vacuum. Solving these two equations, the dispersion relationship for a wave propagating on the surface is

: $k_\left\{x\right\}=frac\left\{omega\right\}\left\{c\right\} left\left(frac\left\{varepsilon_1varepsilon_2\right\}\left\{ varepsilon_1+varepsilon_2\right\} ight\right)^\left\{1/2\right\}.$

In the free electron model of an electron gas, the dielectric constant is [cite book |last=Kittel |first=Charles |authorlink=Charles Kittel |year=1996 |title=Introduction to Solid State Physics |edition= 8th ed. |location= Hoboken, NJ |publisher=John Wiley & Sons |isbn=0-471-41526-X]

: $varepsilon\left(omega\right)=1-frac\left\{omega_\left\{P\right\}^2\right\}\left\{omega^2\right\},$

where the plasma frequency is

: $omega_\left\{P\right\}=sqrt\left\{frac\left\{4pi n e^2\right\}\left\{m^*$

where "n" is the electron density, "e" is the charge of the electron, and "m"* is the effective mass of the electron. The dispersion relationship is plotted in Figure 4. At low k, the surface plasmon behaves like a photon, but as "k" increases, the dispersion relationship bends over and reaches an asymptotic limit. Since the dispersion curve lies to the right of the light curve, $omega=k*c$, the surface plasmon is non-radiative. Finally solving these equations, the maximum frequency of the surface plasmon is calculated to be

: $omega_\left\{SP\right\}=omega_P/sqrt\left\{1+varepsilon_2\right\}.$

In the case of air, this result simplifies to

: $omega_\left\{SP\right\}=omega_P/sqrt\left\{2\right\}.$

If we assume that $varepsilon_2$ is real and $varepsilon_2>0$, then it must be true that $varepsilon_1<0$, a condition which is satisfied in metals. EM waves passing through a metal experience damping due to electrical conductivity and electron-core interactions. These effects show up in as an imaginary component of the dielectric function. The dielectric function of a metal is expressed $varepsilon_1=varepsilon_1\text{'}+i varepsilon_1"$ where $varepsilon_1\text{'}$ and $varepsilon_1"$ are the real and imaginary parts of the dielectric function, respectively. Generally $|varepsilon_1\text{'}| gg varepsilon_1"$ so the wavenumber can be expressed in terms of its real and imaginary components

: $k_\left\{x\right\}=k_\left\{x\right\}\text{'}+i k_\left\{x\right\}"=left \left[frac\left\{omega\right\}\left\{c\right\} left\left( frac\left\{varepsilon_1\text{'} varepsilon_2\right\}\left\{varepsilon_1\text{'} + varepsilon_2\right\} ight\right)^\left\{1/2\right\} ight\right] + i left \left[frac\left\{omega\right\}\left\{c\right\} left\left( frac\left\{varepsilon_1\text{'} varepsilon_2\right\}\left\{varepsilon_1\text{'} + varepsilon_2\right\} ight\right)^\left\{3/2\right\} frac\left\{varepsilon_1"\right\}\left\{2\left(varepsilon_1\text{'}\right)^2\right\} ight\right] .$

The wave vector gives us insight into physically meaningful properties of the electromagnetic wave such as its spatial extent and coupling requirements for wave vector matching.

Propagation length and skin depth

As a surface plasmon propagates along the surface, it quickly loses its energy to the metal due to absorption. The intensity of the surface plasmon decays with the square of the electric field, so at a distance "x", the intensity has decreased by a factor of $exp \left[-2k_\left\{x\right\}"x\right]$. The propagation length is defined as the distance for the surface plasmon to decay by a factor of $1/e$. This condition is satisfied at a lengthcite book |last=Homola |first=Jirí |year=2006 |title=Surface Plasmon Resonance Based Sensors |location=Berlin |publisher=Springer-Verlag |isbn=3-540-33918-3|serie=Springer Series on Chemical Sensors and Biosensors, 4 ]

: $L=frac\left\{1\right\}\left\{2 k_\left\{x\right\}"\right\}.$

Likewise, the electric field falls off exponentially normal to the surface. The skin depth is defined as the distance where the electric field falls off by $1/e$. The field will fall off at different rates in the metal and dielectric medium and the skin depth in each medium can be expressed

: $z_\left\{i\right\}=frac\left\{lambda\right\}\left\{2 pi\right\} left\left(frac$

where $psi$ is the polarization angle and $heta$ is the angle from the "z"-axis in the "xz"-plane. Two important consequences come out of these equations. The first is that if $psi=0$ (s-polarization), then $|W|^2=0$ and the scattered light $frac\left\{dI\right\}\left\{ d Omega I_\left\{0=0$. Secondly, the scattered light has a measurable profile which is readily correlated to the roughness. This topic is treated in greater detail in reference .

Experimental applications

The excitation of surface plasmons is frequently used in an experimental technique known as surface plasmon resonance (SPR). In SPR, the maximum excitation of surface plasmons are detected by monitoring the reflected power from a prism coupler as a function of incident angle or wavelength. This technique can be used to observe nanometer changes in thickness, density fluctuations, or molecular adsorption.

In surface second harmonic generation, the second harmonic signal is proportional to the square of the electric field. The electric field is stronger at the interface because of the surface plasmon. This larger signal is often exploited to produce a stronger second harmonic signal.

ee also

References

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