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
=
(
)