Quantum resistance and conductance

QUANTUM RESISTANCE AND CONDUCTANCE

For an elementary description of quantum conductance effects, it is more appropriate to deal with 1D mesoscopic semiconductor structures like quantum wires.

If the wire is short enough, ie., shorter than the electron mean free path in the material, there will be no scattering and the transport is ballistic..

As in fig. 5.13, the 1D quantum wire is connected through ideal leads. They do not produce scattering events to reservoirs characterized by Fermi levels E_F1 and E_F2

In order to flow of the current through the quantum wire, a small voltage V is applied between the reservoirs.

As a consequence, there is a potential energy eV between the two reservoirs equal to E_F1 - E_F2. The current across the wire should be given by the product of the concentration of (E), electrons (obtained from the density of states function n_1D in the energy interval (eV), the electron velocity v (E), and the unit electronic charge:

I = en_1D (E) v (E) eV … (1)

Substituting for n_1D (E) by its expression, we obtain the following value of the current:

I = 2e² / h V … (2)

It is independent of the carrier velocity.

The value of the conductance G = (I/V) is therefore:

It is interesting to observe that the conductance of the quantum wire is length independent, in contrast to the classical case where it varies inversely to the length.

The quotient

G0 = e² / h  … (4)

is called the quantum unit of conductance. The quantum resistance value is given by

R₀ = h/e² = 25.81 k Ω   … (5)

This can be experimentally determined.

Since the quantity 2e²/h appears very often, it is usually called fundamental conductance.

The above results on quantum conductance and resistance have been derived in the simplest possible manner, using a 1D mesoscopic system.

This quantification of macroscopic classical concepts, like conductance and resistance, is of fundamental importance in mesoscopic physics.

Carbon

In nanotechnology, researchers are looking out for certain materials with desired properties through which the nanoscale components and structures can be obtained.

Carbon is found to be one such material suitable for nano- technology based components due to its inherent desirable properties.

Carbon is a unique atom among other elements because of its ability to exist in a wide variety of structures and forms as shown in the fig. 5.14.

Pure carbon exists in four different crystalline forms namely Diamond, Graphite, Fullerenes and Nanotubes.

Carbon atom is the basic building block of these crystalline structure. Among these, Fullerenes and Nanotubes are found to be useful in nanotechnology for various fabrication of nanostructures.