0.10 Sspd_chapter 1_part 12_quantum mechanical interpretation of  (Page 4/4)

Or J = I/A = (1/ρ)×(V/L)

Therefore V/I = R = (ρL)/A→ Ohm’s Law. ……………………………….. 1.112

In Eq.(1.109) , we find that drift mobility is directly dependent on the mean free time between two consecutive scatterings. Mean free time (τ) is dependent on scattering. Larger is the perturbation in the lattice network from the ideal lattice more frequent will be the scattering and hence shorter will be the mean free time. As the perturbation from the ideal condition or the disorderliness is reduced, so will the scattering phenomena be reduced and mean free time will become longer. Under ideal condition there will be no scattering and mean free time will become infinite. Here the mobility becomes infinite , conductivity becomes infinite and resistivity becomes zero. This is a superconductor. Here once an electron gets an impulsive push it continues to travel in a straight line for infinite distance and for infinite time without any energy dissipation. Hence initial kinetic energy imparted by the impulsive push is conserved forever by the conducting electron. This is tantamount to a current flow in a close loop superconductor without any battery connected to the circuit.

In a normal conductor even with no battery connected, the mobile carriers are undergoing random motion with no net displacement. When an electric field is applied then superimposed on this random motion there is a net displacement of electrons in the opposite direction to the electric field. This has been shown in Figure(1.71). The net displacement is given by ΔL in time Δt.

As can be seen in Figure 1.71 electron is undergoing continuous random motion under the influence of thermal energy. As temperature increases electrons become more restless and start meandering along more zig zag path but they never make a net displacement in any direction as shown by ‘thin line’ zig zag path. But as an electric field(ε) is applied in – z direction, electron follows bolder line and as seen in Figure 1.71 it experiences a net displacement of ΔL in +z direction in Δt.

ΔL/Δt = average drift velocity = v _drift = µ _n ε ;

In 1911, Kamerling Ons detected superconductivity and superfluidity in solid mercury at 4.3Kelvin.

In 1962 a Russian Scientist was awarded the Nobel Prize in Physics for his study on superfluidity and superconductivity in liquid Helium at 4Kelvin.

In 1987 Karl Alex Muller and Bednorz of Germany were awarded the Nobel Prize for discovering the superconductivity in ceramic Yettrium Barium Copper Oxide [Y ₁ Ba ₂ (CuO) ₃ ] at liquid Nitrogen temperature 77Kelvin. Bednorz was a research student under Muller at that time.

Before we leave this Chapter on mobility and resistance it will be appropriate to introduce the reader to the concept of two kinds of mobilties:

First kind μ _lattice due to the scattering caused due to lattice thermal vibration and it is temperature dependent;

Second kind μ _impurity due to the scattering caused due to the dopents and/or crystalline defects;

At liquid Helium temperature that is at less than 4Kelvin there is no scattering due to thermal vibrations but scattering due to impurity and/or crystal defect persist. Therefore semiconductors never become superconductors. Even at 0Kelvin residual resistivity persists due to impurity and/or crystal defect.

So the effective mobility is given by the following reciprocal relationship:

1/ μ =1/ μ lattice + 1/ μ impurity …………………………………….. 1.113

The quantum-mechanical model of electron scattering , which has been presented in this chapter is valid only when drift velocity is much lower than the thermal velocity. When we reach high Electric Field region, the electric energy is directly transferred to the crystalline lattice and the drift velocity saturates at Scatter Limited Velocity as shown in Figure(1.72). The Scatter Limited Velocity is 10 ⁷ cm/sec for Silicon and it can be derived from the following relation:

(1/2)(m e *)(v scatter limited ) 2 = (1/2)(m e *)(v thermal ) 2 = (3/2)kT……………………………. 1.114

Figure 1.72. In high electric field region the drift velocity saturates at scatter limited velocity.

The Quantum Mechanical perspective tells us that electron is not impeded by the lattice centers. If the lattice centers are perfectly orderly at 0Kelvin then electron if imparted an impulsive energy will acquire a finite kinetic energy and with this KE it will continue to travel in a straight line through the crystalline lattice till infinity. What impedes the flow and causes the loss of KE is not lattice centers per se but the disorderliness of the lattice center network. This is the reason why Graphene is working out to be a wonder material with a very large drift mobility. Graphene is a sheet of orderly arrangement of hexagonal structure which is unperturbed or unbroken over large distances hence mean free path in Graphene is of the order of micrometers as compared to the mean free path in GaAs where it is of the order of a fraction of micrometer.