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Updated: Jul 4, 2023

Plasmons: Collective oscillations of the “free electron gas density” often at optical frequencies. The coupling of light to the metal nano particles produces resonances under specific conditions. The resonant interaction between them will localize the electromagnetic field near metal surface and drastically enhances the optical scattering phenomenon. The unique properties of the interface waves result from the frequency-dependent dispersion characteristics of metallic and dielectric materials. Thus, a dielectric–metallic interface can support surface plasmon polaritons (SPP). The momentum of a surface plasmon mode is greater than the momentum of propagation light modes of same frequency. There needs to be an Ex component of the field for a surface charge to exist. Hence SPs yield TM polarized fields. The EM fields of SPs are highly confined to the metal/dielectric interface. They evanescently decrease in strength away from the interface. The tangential and normal components of the SPP wave vector have imaginary parts even in a lossless system. Consequently, the amplitude of such an SPP should decrease in the direction of propagation and increase away from the interface.

Applications of SPP

Highly subwavelength radial miniaturization

Able to concentrate energy in wavelength volume

Strong dispersive properties mechanism for manipulating waves

Able to guide waves along the surface of a metal.

Dispersion of Surface plasmon polaritons

Surface plasmon polaritons (SPPs) are quasiparticles that are formed when light couples with electrons at the interface between a metal and a dielectric material. The dispersion of SPPs refers to the relationship between the frequency and wavevector of the SPPs, and it determines the behavior of the SPPs as they propagate along the interface.

In general, the dispersion of SPPs is described by a dispersion relation, which is a mathematical equation that describes the relationship between the frequency, wavevector, and other material parameters of the system. The dispersion of SPPs is influenced by several factors, including the metal used, the dielectric material, and the thickness of the metal film.

For instance, the dispersion of SPPs can be either positive or negative, depending on the combination of metal and dielectric material used. Positive dispersion means that the frequency of the SPPs increases with increasing wavevector, while negative dispersion means that the frequency decreases with increasing wavevector.

The dispersion of SPPs is important for several applications, such as surface-enhanced spectroscopies, photonics devices, and subwavelength imaging. The understanding of the dispersion of SPPs is crucial for the design and optimization of these applications.

Derivation of the dispersion equation

This relation is the basis for understanding of coupling of light to propagating surface plasmons, by using special approaches to match the wavevector. Surface plasmon polariton is a surface wave propagating along the metal dielectric interface or it is a coupled state between a photon and plasmon.


Incident Wave: E and H depends on Z-variable and time in the form of plane wave

Surface wave : EM wave energy concentrated at the interface (z direction)

and decay away from the interface, Surface ,

are independent on Y




-Amplitudes of Electric and magnetic fields along the x direction

The energy of polaritons is contributed by two parts. One part is pure EM energy accumulated in EM field and second part is energy of solid state.

Light couples with electron oscillations (Plasma oscillations in metal)

Therefore,

EM energy +kinetic energy of free electrons

Energy of resulting wave (SPP)

A plane wave has translational symmetry along z and Y directions

Without loss of generality we can assume that E and H are independent on Y. We need to find distribution of E an M Field.

I.e. Dependence of frequency of surface plasmon polariton on propagation constant kz

Using boundary conditions

We need to find dispersion equation of surface EM wave using from the fundamental Maxwell’s equations Maxwell’s equation

Equivalent to 6 scalar equations

A t the interface six components of electric and magnetic field can be divided into two sets



TM polariton i.e. connection between Hy, Ex, Ez component of EM field


TE polarization, TE polarization- Ey, Hx & Hz

In TM polarization only we can get non-trivial solution

Dispersion Equation


Helmholtz equation for Hy component of magnetic field

Ex and Ez in terms of magnetic field


Solution for Helmholtz equation in upper and lower medium

Upper medium

Lower medium

After applying boundary conditions x=0,

Continuity of tangential component of magnetic field

H1y (0) =H2y (0)

Continuity of tangential component of electric field

Substitute x=0 in the above equations

E1z (0) =E2z (0)

k1, k2 inverse penetration depths in medium1 and medium 2 respectively.

Substitute k1 & k2, k1= k2 =positive

Dispersion equation of SPP

This expression relates the wavenumber of the SPP along the propagation direction with the wavenumber of electromagnetic waves in vacuum k0. The unique properties of the interface waves result from the frequency-dependent dispersion characteristics of metallic and dielectric materials.





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crystgandhi

1. Introduction

Nanotechnology deals with the physiochemical, optical, electrical, and mechanical properties of materials whose size and shape are engineered at the nanoscale. Nanomaterials (NMs) have gained prominence in technological advancements due to their tunable physical, chemical, and biological properties with enhanced performance over their bulk counterparts.


NMs are categorized depending on their size, composition, shape, and origin. The ability to predict the unique properties of NMs increases the value of each classification.


Nano structures refer to materials or objects that have dimensions in the nanometer scale, which is about 1 to 100 nanometers. Nanoscale materials and structures exhibit unique properties that are different from those of their bulk counterparts due to the high surface-to-volume ratio, quantum size effects, and the confinement of electrons in small dimensions.


Nano structures can be made from various materials such as metals, ceramics, polymers, and composites. They can be prepared using various techniques such as top-down (for example, etching and lithography) and bottom-up (for example, self-assembly and chemical synthesis) approaches.


Nano structures have a wide range of applications, including in electronics, energy, biomedical, and environmental fields. For example, they can be used to improve the performance of batteries, create more efficient solar cells, or develop new drug delivery systems.


It's worth noting that while nano structures have the potential to bring significant benefits, they also pose potential health and environmental risks due to their small size and large surface area, which can make them more reactive and easier to penetrate biological membranes. Therefore, it's important to carefully consider the potential benefits and risks of nano structures in their design, synthesis, and use.

The nanomaterials have at least one dimension of 1–100 nm. The nanomaterials exist in single, aggregated, or fused forms with several shapes such as tubular, spherical, and irregular. The most common types of nanomaterials are nanofibers, nanotubes, quantum dots, and nanosheets.


Nanoparticles (NPs)

• Smaller in structures but larger than QDs, usually ranging from 8 to 100 nanometers.

• Exhibit behaviors between those bulk materials and atoms or molecules.

• Possess unexpected optical properties as their size allows for quantum confinement effects.

Low-dimensional nanomaterials with a size of 1–100 nm exhibit the distinctive features responding to their specific characteristics. Their features depend on the synthesis routes (top-down or bottom-up procedures) and the growth methods in solid, liquid, vapor and hybrid phases. Products are finally classified into 0D, 1D, 2D, or 3D dimensional materials based on their sizes in each dimension (x, y, or z) measured in nanoscale size range. This classification is highly dependent on the electron movement along the dimensions in the NMs.


(i) Zero-dimensional nanomaterials: Here, all dimensions (x, y, z) are at nanoscale, i.e., no dimensions are greater than 100 nm. It includes nanospheres and nanoclusters. The electrons are confined in all directions. The electrons are not allowed to move anywhere in the system. The consequence of this confinement in space is the quantization of their energy and momentum. In this case they are subjected to principles of quantum mechanical motion rather than classical mechanics. The motion of electrons in the confined space can be modeled by the motion of a particle in a potential well with infinite walls. The most common representation of zero-dimensional NMs are NPs.

Nanoparticles are of interest because of the new properties (such as chemical reactivity and optical behavior) that they exhibit compared with larger particles of the same materials.

For example, titanium dioxide and zinc oxide become transparent at the nanoscale, however, are able to absorb and reflect UV light, and have found application in sunscreens.


Nanoparticles can also be arranged into layers on surfaces, providing a large surface area and hence enhanced activity, relevant to a range of potential applications such as catalysts.

Fullerene - Composed of at least 60 atoms of carbon, wrapped-up graphene-buckyball

• A cage-like carbon cluster, made up of 12 pentagons and 20 hexagons, five-and six-membered ring patterns.

• C60is the most stable and spherical in shape, diameter of C60 fullerene is about 0.7 nm

C60 fullerene is a type of nano structure that is made of carbon atoms and has a spherical shape. It is one of the simplest and best-studied fullerenes, a family of carbon-based nanostructures that also includes other shapes, such as tubes and ellipsoids. C60 fullerene was first synthesized in 1985, and since then it has been extensively studied for its unique physical and chemical properties.


One of the key properties of C60 fullerene is its high stability and resistance to chemical reactions. This makes it useful as a component in a variety of applications, including as a lubricant, a scavenger of free radicals, and a building block for the synthesis of other nanostructures.


In addition to its stability, C60 fullerene has also been shown to have interesting electronic properties, which make it useful as a material in electronic devices, such as field-effect transistors and photovoltaic cells. It can also act as a host for other species, such as metal atoms and molecules, leading to the formation of functional materials with novel properties. Overall, C60 fullerene is an important example of a nano structure that has been well studied and has numerous potential applications.


Quantum dots

Nanoparticles of semiconductors (quantum dots) were theorized in the 1970s and initially created in the early 1980s. If semiconductor particles are made small enough, quantum effects come into play, which limit the energies at which electrons and holes (the absence of an electron) can exist in the particles. As energy is related to wavelength (or color), this means that the optical properties of the particle can be finely tuned depending on its size. Thus, particles can be made to emit or absorb specific wavelengths (colors) of light, merely by controlling their size.

Recently, quantum dots have found applications in composites, solar cells and fluorescent biological labels (for example to trace a biological molecule) which use both the small particle size and tunable energy levels.

(ii) One-dimensional nanomaterials: Here, two dimensions (x, y) are at nanoscale and the other is outside the nanoscale. This leads to needle shaped nanomaterials. It includes nanofibers, nanotubes, nanorods, and nanowires.


Nanowires

Nanowires are ultrafine wires or linear arrays of dots, formed by self-assembly. They can be made from a wide range of materials. Semiconductor nanowires made of silicon, gallium nitride and indium phosphide have demonstrated remarkable optical, electronic and magnetic characteristics (for example, silica nanowires can bend light around very tight corners).

Nanowires have potential applications in high-density data storage, either as magnetic read heads or as patterned storage media, and electronic and opto-electronic nanodevices, for metallic interconnects of quantum devices and nanodevices.

The preparation of these nanowires relies on sophisticated growth techniques, which include self-assembly processes, where atoms arrange themselves naturally on stepped surfaces, chemical vapor deposition (CVD) onto patterned substrates, electroplating or molecular beam epitaxy (MBE). The ‘molecular beams’ are typically from thermally evaporated elemental sources.


(iii) Two-dimensional nanomaterials: Here, one dimension (x) is at nanoscale and the other two are outside the nanoscale. The 2D nanomaterials exhibit platelike shapes. It includes nanofilms, nanolayers and nano coatings with nanometer thickness. Due to their high anisotropy and chemical functions, two-dimensional (2D) nanomaterials have attracted increasing interest and attention from various scientific fields, including functional electronics, catalysis, supercapacitors, batteries, and energy materials. In the biomedical field, 2D nanomaterials have made significant contributions to the field of nanomedicine, especially in drug/gene delivery systems, multimodal imaging, biosensing, antimicrobial agents and tissue engineering.

2D nanomaterials are considered to be the thinnest nanomaterials due to their thickness and dimensions on macroscale/nanoscale. These nanomaterials have a layered structure with strong in-plane bonds and weak van der Waals (vdW) between layers. These ultrathin nanomaterials can be produced from laminated precursors described in the following sections. Although the ideal state is a single layer, but often these nanosheets are composed of few layers (less than ten layers). In recent years, 2D nanomaterials such as graphene, hexagonal boron nitride (hBN), and metal dichalcogenides (MX2) have attracted a lot of attention due to their satisfactory properties and widespread uses in the electronics, optoelectronics, catalysts, energy storage facilities, sensors, solar cells, lithium batteries, composites, etc.

Inspired by the unique optical and electronic properties of graphene, 2D layered materials – as well as their hybrids – have been intensively investigated in recent years, driven by their potential applications mostly for nanoelectronics.

The broad spectrum of atomic layered crystals includes transition metal dichalcogenides (TMDs), semiconducting dichalcogenides, monoatomic buckled crystals, such as black phosphorous (BP or phosphorene), and diatomic hexagonal boron nitride (h-BN).

This class of materials can be obtained by exfoliation of bulk materials to small scales, or by epitaxial growth and chemical vapor deposition (CVD) for large areas.

Such atomically thin, single- or few-layer crystals are featured with strong intralayer covalent bonding and weak interlayer van der Waals bonding, resulting in superior electrical, optical and mechanical properties.


(iv) Three-dimensional nanomaterials: These are the nanomaterials that are not confined to the nanoscale in any dimension. These materials have three arbitrary dimensions above 100 nm. No dimension at the nanoscale, all dimensions at the macroscale. The bulk (3D) nanomaterials are composed of a multiple arrangement of nano size crystals in different orientations. It includes dispersions of nanoparticles, bundles of nanowires and nanotubes as well as multi nano layers (polycrystals) in which the 0D, 1D and 2D structural elements are in close contact with each other and form interfaces.

This class can contain bulk powders, dispersions of nanoparticles, bundles of nanowires, and nanotubes as well as multi-nanolayers.



These changes arise through systematic transformation in density of electronic energy levels as a function of size, and these changes result in strong variations in the optical and electrical properties with size.

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