Minimum value of conductivity of a Semiconductor:
In previous post Derivation of conductivity of Semiconductor we derived the expression of conductivity of a semiconductor , that is
=
(1)
where
= charge on carrier
= electron concentration per unit volume
= hole concentration per unit volume
= mobility of electron
= mobility of holes
From “Mass Action” law we know that
=
/
(2)
from equation (1) and (2)
=
(
+(
/
)
) (3)
on differentiating equation (3) with respect to
=
(
-(
/
)
) (4)
For maximum or minimum of conductivity ()
put =
from equation (4)
(
-(
/
)
) =
after calculation we will get
=
√(
/
)
Again differentiate equation (4) with respect to to know occurrence of minimum or maximum at
=
√(
/
) .
/
=
/
(5)
After putting the value of =
√(
/
) in equation (5) we will get
/
= a positive quantity
that means at =
√(
/
) minimum conductivity occurs
and minimum value of conductivity () can be calculated by putting the value of
=
√(
/
) in equation (3)
=
√(
)
Try to solve these Important Problems
Q1. Consider a piece of pure silicon 100 µm long with a cross-sectional area of 1 µm2. How much current would flow through this “resistor” at room temperature in response to an applied voltage of 1 volt?
Q2 (a) Show that the minimum conductivity of a semiconductor sample occurs when
=
√(
/
)
(b) What is the expression for the minimum conductivity?
(c) Is this value greatly different than the value of intrinsic conductivity ? given that
= 1.45 x 1010 cm-3,
= 1500 cm2 volt-1 sec-1 and ,
= 500 cm2 volt-1 sec-1,
