Chapter 1 Review of Basic Semiconductor Physics - courses.cit ...

Semiconductor Optoelectronics (Farhan Rana, Cornell University)
Equal energy surfaces in q-space are now spherical shells. We have simply scaled the coordinates.
3
d q
Volume element
3
( 2
)
is q-space corresponds to volume element
3 3
m d k
in k-space.
mxm
y mz
2
Number of states in volume d k
3 in k-space is,
3
d k
2 V 2 V
mxm
ymz
3
d q
3 3
3 2
2
m 2
Number of states in volume d q
3
2 V
mxm
ymz
3 2
m
3
d q
3 2
in q-space is,
Number of states in spherical shell of radius q in q-space is,
2 V
mxmy
mz
3 2
m
2
4
q
3 2 2
dq V
2
m x my
mz
3
E Ec
dE
since
E
2 2
q
Ec
2m
Therefore, the density of states is,
2
gc E 2
mxmymz
3
3
2
E Ec
2 mde
2
2
E Ec
where the density of states effective mass m de for electrons is
1
mde mxmymz3 We can write the conduction band density of states as follows,
3
2 mde
2
gc
E2
2
0
E Ec
for
for
E Ec
E Ec
1.7 Occupation Statistics
1.7.1 The Fermi-Dirac Distribution Function:
The probability of an electron occupying a state of energy E in the crystal is given by the Fermi-Dirac
distribution function,
1
f E EEf KT
1
e
where E f is the Fermi level (or the chemical potential) and K is the Boltzmann constant.
Example:
Consider isotropic parabolic conduction band with the dispersion,
E
k
2 2
k
Ec
2me
q