EMS 172
Profs. Y. Takamura and A. Moulé, 2009
UNIVERSITY OF CALIFORNIA, DAVIS
Department of Chemical Engineering and Materials Science
EMS 172: Electronic, Magnetic, and Optical Properties of Materials
Homework 5  Solutions
Due Date: Tuesday, Nov. 10
th
, 2009 in class
Note: You must show all the steps you used to obtain your answer in order to receive full credit.
1. Extrinsic Semiconductors
[34
pts]
[Computer problem] Consider an
n
type silicon sample that has been doped with 2x10
15
phosphorus atoms per
cm
3
. The donor level for P in Si is 0.045 eV below the conduction band energy.
(a) Use your Matlab / Mathematica program from HW 4 to create plots of
log(N
e
)
(#/m
3
) vs. temperature AND
log(
N
e
)
(#/m
3
) vs. (1/T) *1000 [K
1
] for the full temperature range from 25 K <
T
< 700 K.
Hand sketches will not
be given full credit. Hint for making sure you get the correct magnitude on your plots: if you use ALL SI units for
all the constants (Joules, kg, meters, etc…), you will get an answer for
N
i
/N
e
with units of [#/m
3
].
An ‘easy’
number to remember is that for Si,
N
i
= 10
10
#/cm
3
at 300 K.
As discussed in lecture, the full temperature range is broken down into three regimes,
intrinsic
,
extrinsic
, and
freezeout
, and each of these regimes is described by a separate equation.
In the freezeout regime, the equation is
given by:
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
−
⎟
⎠
⎞
⎜
⎝
⎛
=
=
+
T
k
E
E
N
N
N
N
B
D
C
D
C
D
e
2
)
(
exp
2
1
2
1
Label these three regimes in both of your plots.
When creating your plots, consider the following questions:
(i) What equation governs
N
e
in the intrinsic regime?
(ii) What equation governs
N
e
in the extrinsic regime?
[6 pts for perfect plot]
[1 pt for missing labels]
[1 plot if shape of curve is
correct but magnitude of data
is incorrect]
[1 pt if one of the regions is
missing]
[1 pt if the transition regions
are incorrect]
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Profs. Y. Takamura and A. Moulé, 2009
(b) Using your plot in (a), estimate the lowest temperature above which all the donors are ionized.
You can estimate the temperature above which all the donors are ionized according to when
3
21
2
1
10
2
2
)
(
exp
2
1
−
+
×
=
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
−
⎟
⎠
⎞
⎜
⎝
⎛
=
=
m
N
T
k
E
E
N
N
N
N
D
B
D
C
D
C
D
e
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 Winter '10
 Takamura
 Magnetism, Angular Momentum, Magnetic Field, Y. Takamura, Profs. Y. Takamura, A. Moulé

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