3.32 Sspd_chapter 1_part 12_quantum mechanical interpretation of  (Page 3/4)

1.12.1. Quantum Mechanical basis of Resistance in a conducting Solid.

An ideal crystal with no lattice defects, no dopents and at zero Kelvin temperature behaves like a superconductor.

In a normal metal, resistance is always present.

Let us consider a cylindrical metal of length L cm and A(cm) ² cross-sectional area. A potential difference of V volts is applied across it as shown in Figure(1.70).

The electric field along the longitudinal axis is:

ε = (V/L) Volt/meter …………………………………….. 1.104

Figure 1.70. Electron drift in z direction(longitudinal axis of the cylinder) after the application of the z-axis electric field.

According to Kinematics, the electric field is applied in z direction. This will cause an acceleration in [-z] direction since electron is negative. This acceleration will continue until the electron gets scattered. At the point of scattering all the kinetic energy of the electron is imparted to the lattice and the electron starts anew from zero velocity. Since this is an statistical phenomena hence acceleration time periods are t ₁ , t ₂ , t ₃ , t ₄ ,…… and the terminal velocities are v _t1 , v _t2 , v _t3 , v _t4 ,…………… Acceleration time periods

t ₁ , t ₂ , t ₃ , t ₄ ,…… are the same as the transit periods along the randomly changing straight line segments as defined in Eq.(1.100)

From kinematics: v _tn = u+at _n

where a = acceleration= F/m e * = qε/ m e * = q(V/L)/ m e * ………………………………. 1.105

But u= initial velocity = 0 therefore v tn = at n = (qε/ m e *) (t n )………………………….. 1.106

Therefore the average drift velocity over n ^th acceleration period is:

v n = (v tn + 0)/2 = (qε/ 2m e *) (t n )………………………….. 1.107

Suppose during the flow of current from one end to the other, a given electron undergoes N scattering. This means it undergoes N acceleration periods.

The average drift velocity during N periods of acceleration which occur during the flow of the current is:

v _drift = [∑ v _n ] /N = [∑ (qε/ 2m _e *) (t _n )]/N

Therefore v _drift = (qε/ 2m _e *)[∑(t _n )]/N

v drift = (qε/ 2m e *)τ = (q τ / 2m e *).ε ……………………………… 1.108

where τ = [∑(t _n )]/N= mean free time as defined in Eq.(1.101) and

μ _n = electron mobility = (q τ / 2m _e *)

This relation was used to determine the electron mobility in Table 1.11, Part 11.

Now under low drift velocity condition, drift velocity varies directly as the applied field ε and the constant of proportionality is known as the drift mobility(μ (cm ² /(v-sec)) as shown in Figure 1.72.

v drift = (q τ/ 2m e *) ε =μ n ε ………………………………………. 1.109

Current density =J= number of Coulombs/second

Therefore J=-q×number of electrons flowing through unit cross-sectional area per second.

If we assume that there is no diffusion of mobile carriers and we only have electric drift of the mobile carriers then number of electrons flowing through unit cross-sectional area per second=(1cm ² )(v _drift )(n)

where n= number of conducting electrons per unit (cm) ²

and v _drift = drift velocity of electrons;

Therefore J= (q.v drift .n) Coulombs/(sec-cm 2 ) ………………………………… 1.110

Or J = q.µ.ε.n =σ.ε…………………………………………………….. 1.111

Where σ(conductivity) = 1/ρ(resistivity)

J = q.µ.ε.n =σ.ε