# Surface plasmon

Schematic representation of an electron density wave propagating along a metal – dielectric interface. The charge density oscillations and associated electromagnetic fields are called surface plasmon-polariton waves. The exponential dependence of the electromagnetic field intensity on the distance away from the interface is shown on the right. These waves can be excited very efficiently with light in the visible range of the electromagnetic spectrum.

Surface plasmons (SPs) are coherent electron oscillations that exist at the interface between any two materials where the real part of the dielectric function changes sign across the interface (e.g. a metal-dielectric interface, such as a metal sheet in air). SPs have lower energy than bulk (or volume) plasmons which quantise the longitudinal electron oscillations about positive ion cores within the bulk of an electron gas (or plasma). When SPs couple with a photon, the resulting hybridised excitation is called a surface plasmon polariton (SPP). This SPP can propagate along the surface of a metal until energy is lost either via absorption in the metal or radiation into free-space. The existence of surface plasmons was first predicted in 1957 by Rufus Ritchie.[1] In the following two decades, surface plasmons were extensively studied by many scientists, the foremost of whom were T. Turbadar in the 1950s and 1960s, and Heinz Raether, E. Kretschmann, and A. Otto in the 1960s and 1970s. Information transfer in nanoscale structures, similar to photonics, by means of surface plasmons, is referred to as plasmonics.[2]

## Excitation

Figure 1: (a) Kretschmann and (b) Otto configuration of an Attenuated Total Reflection setup for coupling surface plasmons. In both cases, the surface plasmon propagates along the metal/dielectric interface
Figure 2: Grating Coupler for Surface Plasmons. The wave vector is increased by the spatial frequency

Surface plasmons can be excited by both electrons and photons. Excitation by electrons is created by firing electrons into the bulk of a metal. As the electrons scatter, energy is transferred into the bulk plasma. The component of the scattering vector parallel to the surface results in the formation of a surface plasmon.[3]

Coupling of photons into SPs can be achieved using 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 parallel wave vector component by an amount related to the grating period (Figure 2). This method, while less frequently utilized, is critical to the theoretical understanding of the effect of surface roughness. Moreover, simple isolated surface defects such as a groove, a slit or a corrugation on an otherwise planar surface provides a mechanism by which free-space radiation and SPs can exchange energy and hence couple.

## Dispersion relation

Figure 3: Coordinate system for 2 material interface

The electric field of a propagating electromagnetic wave can be expressed

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

where k is the wave number and ω is the frequency of the wave. By solving Maxwell's equations for the electromagnetic wave at an interface between two materials with relative dielectric functions ε1 and ε2 (see figure 3) with the appropriate continuity relation the boundary conditions are[4][5]

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

and

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

where c is the speed of light in a vacuum, and kx is same for both media at the interface for a surface wave. Solving these two equations, the dispersion relation for a wave propagating on the surface is

$k_{x}=\frac{\omega}{c} \left(\frac{\varepsilon_1\varepsilon_2}{ \varepsilon_1+\varepsilon_2}\right)^{1/2}.$
Figure 4: Dispersion curve for surface plasmons. At low k, the surface plasmon curve (red) approaches the photon curve (blue)

In the free electron model of an electron gas, which neglects attenuation, the metallic dielectric function is[6]

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

where the bulk plasma frequency in SI units is

$\omega_{P}=\sqrt{\frac{n e^2}{{\varepsilon_0}m^*}}$

where n is the electron density, e is the charge of the electron, m* is the effective mass of the electron and ${\varepsilon_0}$ is the permittivity of free-space. The dispersion relation is plotted in Figure 4. At low k, the SP behaves like a photon, but as k increases, the dispersion relation bends over and reaches an asymptotic limit corresponding to the surface plasma frequency. Since the dispersion curve lies to the right of the light line, ω = k·c, the SP has a shorter wavelength than free-space radiation such that the out-of-plane component of the SP is purely imaginary and exhibits evanescent decay. The surface plasma frequency is given by

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

In the case of air, this result simplifies to

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

If we assume that ε2 is real and ε2 > 0, then it must be true that ε1 < 0, a condition which is satisfied in metals. Electromagnetic waves passing through a metal experience damping due to Ohmic losses 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 ε1 = ε1' + i·ε1" where ε1' and ε1" are the real and imaginary parts of the dielectric function, respectively. Generally |ε1'| >> ε1" so the wavenumber can be expressed in terms of its real and imaginary components as[4]

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

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 an SPP propagates along the surface, it loses 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[-2kx"x]. The propagation length is defined as the distance for the SPP intensity to decay by a factor of 1/e. This condition is satisfied at a length[7]

$L=\frac{1}{2 k_{x}''}.$

Likewise, the electric field falls off evanescently perpendicular to the metal surface. At low frequencies, the SPP penetration depth into the metal is commonly approximated using the skin depth formula. In the dielectric, the field will fall off far more slowly. The decay lengths in the metal and dielectric medium can be expressed as[7]

$z_{i}=\frac{\lambda}{2 \pi} \left(\frac{|\varepsilon_1'|+\varepsilon_2}{\varepsilon_i^2} \right)^{1/2}$

where i indicates the medium of propagation. Surface plasmons are very sensitive to slight perturbations within the skin depth and because of this, surface plasmons are often used to probe inhomogeneities of a surface.

## Effects of roughness

In order to understand the effect of roughness on surface plasmons, it is beneficial to first understand how a plasmon is coupled by a grating Figure2. When a photon is incident on a surface, the wave vector of the photon in the dielectric material is smaller than that of the SPP. In order for the photon to couple into a SPP, the wave vector must increase by $\Delta k = k_{SP}- k_{x, \text{photon}}$. The grating harmonics of a periodic grating provide additional momentum parallel to the supporting interface to match the terms.

$k_{SP}=k_{x, \text{photon}} \pm n\ k_\text{grating}=\frac{\omega}{c} \sin{\theta_0} \pm n \frac{2\pi}{a}$

where $k_\text{grating}$ is the wave vector of the grating, $\theta_0$ is the angle of incidence of the incoming photon, a is the grating period, and n is an integer.

Rough surfaces can be thought of as the superposition of many gratings of different periodicities. Kretschmann proposed[8] that a statistical correlation function be defined for a rough surface

$G(x,y)=\frac{1}{A}\int_A z(x',y')\ z(x'-x,y'-y)\, dx'\, dy'$

where $z(x,y)$ is the height above the mean surface height at the position $(x,y)$, and $A$ is the area of integration. Assuming that the statistical correlation function is Gaussian of the form

$G(x,y)=\delta^2\exp\left(-\frac{r^2}{\sigma^2}\right)$

where $\delta$ is the root mean square height, $r$ is the distance from the point $(x,y)$, and $\sigma$ is the correlation length, then the Fourier transform of the correlation function is

$|s(k_\text{surf})|^2=\frac{1}{4 \pi} \sigma^2 \delta^2 \exp \left( - \frac{\sigma^2 k_\text{surf}^2}{4}\right)$

where $s$ is a measure of the amount of each spatial frequency $k_\text{surf}$ which help couple photons into a surface plasmon.

If the surface only has one Fourier component of roughness (i.e. the surface profile is sinusoidal), then the $s$ is discrete and exists only at $k=\frac{2\pi}{a}$, resulting in a single narrow set of angles for coupling. If the surface contains many Fourier components, then coupling becomes possible at multiple angles. For a random surface, $s$ becomes continuous and the range of coupling angles broadens.

As stated earlier, surface plasmons are non-radiative. When a surface plasmon travels along a rough surface, it usually becomes radiative due to scattering. The Surface Scattering Theory of light suggests that the scattered intensity $dI$ per solid angle $d \Omega$ per incident intensity $I_{0}$ is[9]

$\frac{dI}{ d \Omega\ I_{0}}=\frac{4 \sqrt{\varepsilon_{0}}}{\cos{\theta_0}}\frac{\pi^4}{\lambda^4}|t_{012}^p|^2 \ |W|^2 |s(k_\text{surf})|^2$

where $|W|^2$ is the radiation pattern from a single dipole at the metal/dielectric interface. If surface plasmons are excited in the Kretschmann geometry and the scattered light is observed in the plane of incidence (Fig. 4), then the dipole function becomes

$|W|^2=A(\theta,|\varepsilon_{1}|)\ \sin^2{\psi} \ [(1+\sin^2 \theta /|\varepsilon_1|)^{1/2} - \sin{\theta}]^2$

with

$A(\theta,|\varepsilon_1|) = \frac{|\varepsilon_1|+1}{|\varepsilon_1|-1} \frac{4}{1+\tan{\theta}/| \varepsilon_1|}$

where $\psi$ is the polarization angle and $\theta$ 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 $\frac{dI}{ d \Omega\ I_{0}}=0$. Secondly, the scattered light has a measurable profile which is readily correlated to the roughness. This topic is treated in greater detail in reference.[9]

## 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 absorption.

Surface plasmon-based circuits have been proposed as a means of overcoming the size limitations of photonic circuits for use in high performance data processing nano devices.[10]

The ability to dynamically control the plasmonic properties of materials in these nano-devices is key to their development. A new approach that uses plasmon-plasmon interactions has been demonstrated recently. Here the bulk plasmon resonance is induced or suppressed to manipulate the propagation of light.[11] This approach has been shown to have a high potential for nanoscale light manipulation and the development of a fully CMOS- compatible electro-optical plasmonic modulator.

CMOS compatible electro-optic plasmonic modulators will be key components in chip-scale photonic circuits.[12]

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 resulting in a non-linear optical effect. This larger signal is often exploited to produce a stronger second harmonic signal.

The wavelength and intensity of the plasmon-related absorption and emission peaks are affected by molecular adsorption that can be used in molecular sensors. For example, a fully operational prototype device detecting casein in milk has been fabricated. The device is based on monitoring changes in plasmon-related absorption of light by a gold layer.[13]

## References

1. ^ Ritchie, R. H. (June 1957). "Plasma Losses by Fast Electrons in Thin Films". Physical Review 106 (5): 874–881. Bibcode:1957PhRv..106..874R. doi:10.1103/PhysRev.106.874.
2. ^ Polman, Albert; Harry A. Atwater (2005). "Plasmonics: optics at the nanoscale". Materials Today. 8 (2005): 56. doi:10.1016/S1369-7021(04)00685-6. Retrieved January 26, 2011.
3. ^ S.Zeng et al.; Yu, Xia; Law, Wing-Cheung; Zhang, Yating; Hu, Rui; Dinh, Xuan-Quyen; Ho, Ho-Pui; Yong, Ken-Tye (2012). "Size dependence of Au NP-enhanced surface plasmon resonance based on differential phase measurement". Sensors and Actuators B: Chemical 176: 1128. doi:10.1016/j.snb.2012.09.073.
4. ^ a b Raether, Heinz (1988). Surface Plasmons on Smooth and Rough Surfaces and on Gratings. Springer Tracts in Modern Physics 111. New York: Springer-Verlag. ISBN 3540173633.
5. ^ Cottam, Michael G. (1989). Introduction to Surface and Superlattice Excitations. New York: Cambridge University Press. ISBN 0750305886.
6. ^ Kittel, Charles (1996). Introduction to Solid State Physics (8th ed.). Hoboken, NJ: John Wiley & Sons. ISBN 0-471-41526-X.
7. ^ a b Homola, Jirí (2006). Surface Plasmon Resonance Based Sensors. Springer Series on Chemical Sensors and Biosensors, 4. Berlin: Springer-Verlag. ISBN 3-540-33918-3.
8. ^ (German) Kretschmann, E. (April 1974). "Die Bestimmung der Oberflächenrauhigkeit dünner Schichten durch Messung der Winkelabhängigkeit der Streustrahlung von Oberflächenplasmaschwingungen". Optics Communications 10 (4): 353–356. Bibcode:1974OptCo..10..353K. doi:10.1016/0030-4018(74)90362-9.
9. ^ a b Kretschmann, E. (1972). "The angular dependence and the polarisation of light emitted by surface plasmons on metals due to roughness". Optics Communications 5 (5): 331–336. Bibcode:1972OptCo...5..331K. doi:10.1016/0030-4018(72)90026-0.
10. ^ Ozbay, E. (2006). "Plasmonics: Merging Photonics and Electronics at Nanoscale Dimensions". Science 311 (5758): 189–93. doi:10.1126/science.1114849. PMID 16410515.
11. ^ Akimov, Yu A; Chu, H S (2012). "Plasmon–plasmon interaction: Controlling light at nanoscale". Nanotechnology 23 (44): 444004. doi:10.1088/0957-4484/23/44/444004. PMID 23080049.
12. ^ Wenshan Cai, Justin S. White, and Mark L. Brongersma (2009). "Compact, High-Speed and Power-Efficient Electrooptic Plasmonic Modulators". Nano Letters 9 (12): 4403–11. doi:10.1021/nl902701b. PMID 19827771.
13. ^ Minh Hiep, Ha; Endo, Tatsuro; Kerman, Kagan; Chikae, Miyuki; Kim, Do-Kyun; Yamamura, Shohei; Takamura, Yuzuru; Tamiya, Eiichi (2007). "A localized surface plasmon resonance based immunosensor for the detection of casein in milk". Science and Technology of Advanced Materials (free download pdf) 8 (4): 331. Bibcode:2007STAdM...8..331M. doi:10.1016/j.stam.2006.12.010.