# Characteristic Lengths In Mesoscopic Systems

## Characteristic Lengths in Mesoscopic Systems

It deals with structures having size between macroscopic and microscopic objects usually varying from a few nanometers to 100nm. Electrons in such systems show their wave like properties and are depended on geometry of the sample. So electrons act as EM radiation in waveguides. It is convenient to classify such system based on their characteristic length as one object may show different properties with respect to the length considered.

Some most commonly used lengths are,

1. De Broglie Wavelength
2. Mean free path
3. Diffusion Length
4. Screening length
5. Localization length

### 1. De Broglie Wavelength in Nanoelectronics

For an electron of momentum P, there exist a wavelength given by,
λB = h/p = h/mv

The mass ‘M’ of an electron is probably considered as vaccum. Once they behave dynamically, their mass is considered as M* which is the effective mass.

The smaller the value of m*, easier will be to observe the size of quantum effect in given nanostructure.

The present lithographic technique makes it easier to construct semiconductor nanostructure with one or two dimension of the order of λB.

### 2. Mean Free Path in Nanoelectronics

Scattering of particle inside a solid occurs due to imperfection like impurity, lattices vibration defects. The distance covered by the electron between two inelastic collision is called the mean free path.

le = vτe
le – distance
v- speed
τe = Mean free time

### 3. Diffusion Length in Nanoelectronics

Electrons can move either in ballistic nature or diffusion nature of a particle move throughout the structure without scattering it is said to have ballistic nature.

Hot electrons transistor exhibits ballistic nature of electron transport.

Consider Diffusion length Le,where, Le very much greater than l.

Diffusion coefficient is represented by D.

Le = (Dτe)1/2

Diffusion is the process of movement of electrons from higher concentration region to lower concentration region. The electron transport is explained by Boltzmann diffusion. For ballistic region, Boltzmann transport region is not valid.

### 4. Screening Length in Nanoelectronics

In extrinsic semiconductor impurities are normally ionised and are considered a main factor for scattering. Due to screening of free carrier by charge of opposite polarity, we cannot consider electrical potential (v) by this impurity to vary with respect to 1/r.

Thus effect of impurity over distance is partially reduced. The variation of potential is dependent on the term, exp [-r/λs]

Where λs = [ϵKt/e2n]

e – electronic charge
ϵ - dielectric constant
n – background carrier concentration
k = Boltzmann constant
T – temperature

λs is in the range of 10 – 100 nm and is the indication of disturbance in the semiconductor. λs should be much smaller in metal than in semiconductor.

Φsp = r[e-r/λs/r]

As λs turns to infinity screening effects disappears. It can also be observed that di-electric function ϵ is distance dependent.

Coulombic length = 1/r
As λs = α
e = α = 1
Φsp = 1/r = coulombic length

### 5. Localisation Length in Nanoelectronics

For transport of electron in disordered material, there can be localised states. From which electron jumps between localised states to other localised state, until it reaches a bounded state.
In order to describe the hopping transport, we can consider the wave function,

Ψ = exp(-r/λloc)
λloc – localisation length.

We use the concept of localised state quantum hall effect.