Derivation of conductivity of Semiconductor

Derivation of conductivity of Semiconductor:

In last post Conductivity in Semiconductor , we knew that flow of current in any semiconductor is due to both electron and holes and we also knew conductivity of semiconductors 

in above figure let current i is flowing in x direction in semiconductor ( cylindrical form ) , this current is addition of two current , current  due to flow of holes and current  due to flow of electron

so  

first we will calculate conductivity due to flow electron only

E = electric field vector across the small  section ( in cylindrical form ) of semiconductors

N= number of electron in above semiconductor

A= cross sectional area

L= length of section of cylindrical semiconductor

let one electron contains charge q than total charge flow will be = Nq

so current flow due to electron will be

 = (Total charge flow)/( time taken in flow of charge)

 =                                      (1)

time  = (Average distance covered by electron) / (drift velocity of electron )

  =                                      (2)

From above equation (1) and (2)

  =                                 (3)

Let conduction current density due to flow of current ( due to flow of electron ) is  , and we know that

 =                                        (4)

From above equation (3) and (4)

=   

 =                                     (5)

where   =   = number of electron per unit volume

Drift velocity of electron can be expressed as

 =                                        (6)

where  is mobility of electron

From Equation (5) and (6)

 =                                     (7)

from ohms low we know that

 =                                           (8)

where  is conductivity due to flow of  electron

by comparing equation (7) with equation (8) , we can concluded that

=                                         (9)

so equation (9) gives the expression of conductivity of semiconductor due to flow of electrons

In similar fashion we can calculate conductivity of semiconductor due to flow of holes

let  holes per unit volume flow in semiconductor bar above shown , and  is the mobility of holes ,  is charge on hole,

so conductivity expression due to holes can be given by

=                                            (10)

So total conductivity  of semiconductor (due to flow of electron as well as holes) can be given by

 = +

=  + 

 = 

For intrinsic semiconductor

where  is the intrinsic carrier concentration per unit volume of intrinsic semiconductor 

so conductivity of intrinsic semiconductor can be given by

 = ()