Problem I
Calculate the magnitude of the velocity of an electron in GaAs with energy of 3/2 k
B
T
above
the
conduction band edge (the thermal average energy) with T = 300K, assuming the validity of the
effective mass approximation. Repeat for Si, treating the system as isotropic (not directionally
dependent) using the conductivity, effective mass. (This actually works.) In each case, put your
answers in units of m/s = nm/ns (relevant for current device scales and clock frequencies) and
km/hour (relevant to everyday experience). Appendix III may be helpful. Note that the
conductivity effective masses are provided in Example 36 of the text, or could be calculated for
Si as done in the example. (Also, just FYI for now, note that the average velocity of charge
carriers through conventional transistors now approaches this velocity.)
Solution:
The effective mass approximation means that the electrons can be treat as “free”
particle inside the semiconductor, if the mass is the effective mass. In another word,
the effect of crystal potentials are all contained in the E(k) relationship, therefore
effective mass.
For GaAs:
*2
23
56
*3
1
31
22
3
3 1.38 10
300
/
4.5 10
/
1.6 10
/
0.067 9.1 10
BG
a
A
s
B
GaAs
kT
m
v
v
nmn
s
s
kmh
r
m
For Silicon:
,
23
55
1
,
3
3 1.38 10
300
/
2.3 10
/
8.2 10
/
0.26 9.1 10
B
Si cond
B
Si cond
m
v
v
s
s
r
m
Problem II
Consider conductionband minimum energy valleys centered at the X points, and assume that for
the valley centered at the X point along the (100) direction (
2/
kx
a
), specifically, the
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 Fall '08
 Banjeree
 effective mass, effective mass approximation

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