Page 188 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Electron Distribution Functions
β
however, that the temperature, which gives us the parameter , enters
into the term for the probability of finding an electron with an energy E
above the conduction band edge energy E , and is given by
C
1
f E() = --------------------------- (5.36)
e
(
e β E – µ) + 1
An intrinsic semiconductor is supposed to be uncharged. Therefore, we
find that with a certain probability f E() = 1 – fE() an electron is
h
missing in the valence band. This missing electron we call a hole or
defect electron. It behaves like a carrier with opposite charge. If the num-
ber densities n of positive (holes) and n of negative (electrons) carriers
h e
are equal the semiconductor is charge free. This number density is given
by the integral over all possible implementations in energy space

∞
d
n = ∫ D E() f E() E (5.37a)
e e e
E c
∞ ∞
(
d
n = ∫ D E() f E() E = ∫ D E() 1 – f E()) E (5.37b)
d
h
h
e
h
h
E V E V
Because of the discussion that lead to (5.8) the density of states for the
valence band (D h ) and for the conduction band (D e ) appears in the inte-
gral. The energy of the valence band edge E V and the conduction band
edge E C is fixed. Their difference E G = E – E V , the band gap energy,
C
is a typical material parameter. The only parameter left to fulfil n = n h
e
µ
is the chemical potential . In the intrinsic case this is usually several
k T below the conduction band, so that
B
 µ – E 
fE() = exp ------------- (5.38a)
 k T 
B
 E – µ 
1 – fE() = exp ------------- (5.38b)
 k T 
B

Semiconductors for Micro and Nanosystem Technology 185