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ECE 440
Homework IV
Fall 2008
Due: Monday, September 22, 2008
1.
In practice we assume that the intrinsic Fermi level, E
i
,
coincides with the center of the band
gap. In reality it is not true. Derive an expression relating the intrinsic level E
i
to the center
of the band gap. Find the displacement of E
i
from the center of the band gap for both silicon
and germanium at room temperature. The effective mass values are m
n
* = 0.55 m
o
for Ge
and 1.1 m
o
for Si; m
p
* = 0.37 m
o
for Ge and 0.56m
o
for Si where m
o
is the free electron
mass.
We know that at thermal equilibrium, the intrinsic carrier concentrations for electrons and
holes are equal. By setting equations 315 & 319 equal to one another, a relationship
between the intrinsic Fermi Level and temperature can be found.
At this stage, it is a good idea to draw an energy band diagram to visualize what these
quantities physically mean.
What we are saying here is that our intrinsic Fermi Level is simply half the energy band gap
plus some offset term. You could have also defined the intrinsic Fermi Level as half the band
gap minus some energy offset as well. Thus we have:
Also, if we use the common convention that Ev = 0, we can say that:
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View Full DocumentBy combining these three equations (plus some simplification) we can find an expression for
the displacement energy with respect to the middle of the gap.
Now we have to plug and chug to find the actual displacements for both Ge and Si at room
temperature (T=300K):
Clearly, the “real” intrinsic Fermi level lies below midgap for both Ge and Si at room
temperature.
2. The electron concentration in a silicon material is 8x10
2
/cm
3
at room temperature under
equilibrium conditions. (a) What is the hole concentration? (b) Where is E
F
positioned
relative to E
i
? (c) Draw the energy band diagram for the material. Repeat for parts (a), (b)
and (c) for the same sample if the temperature is 400 K.
Refer to Fig. 317 to obtain the
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