##
**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,

λ

_{B}= 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 λ

_{B.}### 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.

l

_{e}= vτ_{e}
l

_{e}– distance
v- speed

τ

_{e}= 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 = (Dτ

_{e})^{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/λ

_{s}]
Where λ

_{s }= [ϵKt/e^{2}n]
e –
electronic charge

ϵ - dielectric
constant

n –
background carrier concentration

k =
Boltzmann constant

T –
temperature

λ

_{s }is in the range of 10 – 100 nm and is the indication of disturbance in the semiconductor. λ_{s}should be much smaller in metal than in semiconductor.
As λ

_{s}turns to infinity screening effects disappears. It can also be observed that di-electric function ϵ is distance dependent.
Coulombic length
= 1/r

As λ

_{s}= α
e = α = 1

Φ

_{sp}= 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/λ

_{loc})
λ

_{loc}– localisation length.
We use the concept of
localised state quantum hall effect.

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