1.1)Electron As A Particle - Conductivity and Ohm's Law

                                           Introduction

 We know that the electron can be regarded as a negative point charge and it obeys the laws of mechanics and electromagnetism. But the properties of a transistor cannot be explained from the above models. Matter consists of atoms in a certain spatial arrangement. This science is known as crystallography. For this part we shall assume that all materials crystallize in simple cubic structures in 3-D.This will help us explain the Ohm’s Law the Hall Effect and several other important events.

A Simple Cubic Cell

It is obvious that the amount of electrons in a neutral solid is equal to the amount of the positive charge in the solid. Secondly we assume that electrons bounce around in the inter-atomic space, colliding occasionally with lattice atoms much like gas molecules. In analogy with the Maxwell-Boltzmann distribution we assume that at equilibrium the electrons follow same statistical distribution as gas molecules, which depends on temperature. Then we may say that the mean thermal  velocity of electrons is given by the formula:

           ½mv_th²  = 3/2k_BT     (an electron in 3-D has 3 degrees of freedom)     ……(1.1)

Later we shall see that the above assumption is not true for atoms, but it is nearly true for conduction electrons in semiconductors.

 The Effect of an Electric Field – Conductivity and Ohm’s Law

Suppose we apply a potential V across two ends of a solid of length L, then the Electric Field at every point in the solid is given by:

                                                        E = V/L                                             …..(1.2)

And produces an acceleration :

                                                       a = eE/m                                       ……(1.3)

in the electron. Thus the electron acquires a velocity in the direction of the electric field. We may assume that the electron loses this directed velocity is completely lost after each collision, because an electron is much lighter than a lattice atom. If τ is the average time between two collisions, the average velocity v_average should be:

                 ( ∫adt)/(∫dt) = 1/2aτ     (integration is done between the limits 0 and τ )            ……(1.4)

But since we have not taken the actual time between collision and then taken the average, the above expression is not correct. A more advanced and a rigorous mathematical treatment gives a factor of 2. Therefore the average velocity is given by:

                                                    v_average = aτ                                         …….(1.5)

This velocity is referred to as the mean velocity or drift velocity. The drift velocity ν_D and τ can be related as:

                                                   ν_D =(eτ/m)E                                      ……..(1.6)

The proportionality constant in the parenthesis is called the mobility.

Assume that all electrons drift with ν_D. The total number of electrons crossing a plane of unit area per second is obtained by multiplying ν_D with the electron density N_e. Multiplying further by the charge on electron we obtain the electric current density:

                                                   J = N_eeν_D                                                     …….(1.7)

Only the drift velocity created by the electric field comes in the expression.

 Combining (1.6) and (1.7) we get:

                                                 J = (N_ee²τ/m)/E                                   …… (1.8)

This linear relation can be recognized as the Ohm’s Law:

                                                       J = σE                                          ….. (1.9)

Where σ is the electric conductivity. Combining (1.8) and (1.9) and get:

                                                 σ = (eτ/m)(N_ee)

                                                    =    μ_e (N_ee)                                  ........(1.10)

Thus we have high conductivity because there are lots of electrons around or because they acquire high drift velocities by having high mobilities(μ_e).Ohm’s Law implies that σ is a constant which means that τ is independent of electric field.From our it is reasonable to assume that l the distance between collisions is constant in the regularly spaced lattice, rather than τ. l and τ can be related as:

                                              l = (v_th + ν_D)τ                  ……(1.11)

But since ν_D varies with Electric Field τ must also vary unless

                                                 V_th ≫ ν_D                       …….(1.12)

and from (1.1) it can be calculated that v_th is of the order of 10⁵ms^-1.Thus making τ independent of the electric field E.