1
3.155J/6.152J
Microelectronic Processing
Fall Term,2003
Bob O'Handley
Martin Schmidt
Problem set 2 Solutions Out Sept,17,2003 Due Sept.24,2003
Problem 1
Deal-Grove Model states that the thickness of the oxide is related to a time constant t and
two constants A and B by the relation
x
ox
=
-A + A
2
+ 4B(t + t)
2
where
t =
x
0
2
+ Ax
0
B
A,B,and t can be obtained from the oxidation of Si lecture notes,For the initial 20
minutes of dry oxidation at 1000°C,
A = 0.165mm
B = 0.0117mm
2
/hr
t = 0.37h
Solving for t
ox
:
x
ox
= 0.0401mm
For the 90 minutes of wet oxidation,A = 0.226 μm and B = 0.287 μm
2
/hr,We need to
calculate a new t using the xox calculated in the previous step and the new A and B values
that apply for the wet oxidation:
t = 0.037 h
After 90 minutes of wet oxidation,
x
ox
= 0.561 mm
The final 20 minutes of dry oxidation has the same A and B as the first step and we
need to calculate a new t from the total xox grown in the previous two steps,0.561 μm.
t = 34.8 h
Solving for our final x
ox
:
2
x = 0.564 m
Using the charts,one can estimate that the oxide thickness should be around 0:56 mm.
This shows that the Deal-Grove model is a good approximation for oxide growth.
Problem 2
The Deal-Grove coefficients for this problem are the same as in problem 1.
A = 0.165mm
B = 0.0117mm
2
t = 0.37h
Using the relation
x
ox
=
-A+A
2
+4B(t+t)
2
After 1 hour,the oxide thickness is:
x
ox
= 0.0686 m
Problem 3
In general,Using the knowledge that you have regarding the oxidation process,explain
qualitatively what you expect to happen when polycrystalline silicon is oxidized and also
what is anticipated when Boron is present,You may use some assumptions about B and
A,but also speculate physically from what was discussed in the oxidation lectures.
How will oxide growth rate be affected by poly versus single-crystal silicon?
To the first order,the same set of models that model single crystal silicon oxidation can
be used for polycrystalline silicon,Polycrystalline means that there are a variety of
crystal orientations present in the grains,which will clearly have an effect on the
parameters of the Deal-Grove model,No crystal orientation effects are expected on the
coefficient B because it describes oxidant diffusion through the SiO2,The ratio B/A is
effected because it involves reaction at the Si/SiO2 interface,For polycrystalline silicon,
Plummer says to use an average B/A value,The average B/A value will be for the rate of
oxidation of the <110> orientation,which is between the two extremes of <100> and
<111>,Orientation effects incorporated as stated in equation 6.40 of Plummer.
B
A
ê
á
ˉ
111
=1.68
B
A
ê
á
ˉ
100
How will oxide growth rate be affected by boron atoms?
Highly doped substrates oxidize more rapidly than do lightly doped wafers,At lower
temperatures,and for thinner oxides,the enhancement can be three to four times that for
undoped Si.
3
B
A
=
B
A
ê
á
ˉ
i
1+ 2.62 ￥10
3
exp -
1.1eV
kT
ê
á
ˉ
C
V
T
C
V
i
T
-1
ê
á
á
ˉ
è
í
í
í
˙
˙
˙
The boron dopant concentration is implied by the usage of the term,heavily-doped”.
Heavily-doped means degenerately-doped,which is doping greater than 10
19
cm
-3
(page
22 in Plummer).
This equation and definition are provided for reference,but to solve this problem,we're
not expecting you to use the equation,Instead,using a physical understanding of the
oxidation process,calculate the oxide thickness for typical silicon material and then
acknowledge that this is a lower bound,and then qualitatively explain how the
polycrystalline nature and doping affect this bound,One could also look at the lecture
notes that show how oxidation thickness is affected by doping and also how B/A and B
are affected.
3.155J/6.152J
Microelectronic Processing
Fall Term,2003
Bob O'Handley
Martin Schmidt
Problem set 2 Solutions Out Sept,17,2003 Due Sept.24,2003
Problem 1
Deal-Grove Model states that the thickness of the oxide is related to a time constant t and
two constants A and B by the relation
x
ox
=
-A + A
2
+ 4B(t + t)
2
where
t =
x
0
2
+ Ax
0
B
A,B,and t can be obtained from the oxidation of Si lecture notes,For the initial 20
minutes of dry oxidation at 1000°C,
A = 0.165mm
B = 0.0117mm
2
/hr
t = 0.37h
Solving for t
ox
:
x
ox
= 0.0401mm
For the 90 minutes of wet oxidation,A = 0.226 μm and B = 0.287 μm
2
/hr,We need to
calculate a new t using the xox calculated in the previous step and the new A and B values
that apply for the wet oxidation:
t = 0.037 h
After 90 minutes of wet oxidation,
x
ox
= 0.561 mm
The final 20 minutes of dry oxidation has the same A and B as the first step and we
need to calculate a new t from the total xox grown in the previous two steps,0.561 μm.
t = 34.8 h
Solving for our final x
ox
:
2
x = 0.564 m
Using the charts,one can estimate that the oxide thickness should be around 0:56 mm.
This shows that the Deal-Grove model is a good approximation for oxide growth.
Problem 2
The Deal-Grove coefficients for this problem are the same as in problem 1.
A = 0.165mm
B = 0.0117mm
2
t = 0.37h
Using the relation
x
ox
=
-A+A
2
+4B(t+t)
2
After 1 hour,the oxide thickness is:
x
ox
= 0.0686 m
Problem 3
In general,Using the knowledge that you have regarding the oxidation process,explain
qualitatively what you expect to happen when polycrystalline silicon is oxidized and also
what is anticipated when Boron is present,You may use some assumptions about B and
A,but also speculate physically from what was discussed in the oxidation lectures.
How will oxide growth rate be affected by poly versus single-crystal silicon?
To the first order,the same set of models that model single crystal silicon oxidation can
be used for polycrystalline silicon,Polycrystalline means that there are a variety of
crystal orientations present in the grains,which will clearly have an effect on the
parameters of the Deal-Grove model,No crystal orientation effects are expected on the
coefficient B because it describes oxidant diffusion through the SiO2,The ratio B/A is
effected because it involves reaction at the Si/SiO2 interface,For polycrystalline silicon,
Plummer says to use an average B/A value,The average B/A value will be for the rate of
oxidation of the <110> orientation,which is between the two extremes of <100> and
<111>,Orientation effects incorporated as stated in equation 6.40 of Plummer.
B
A
ê
á
ˉ
111
=1.68
B
A
ê
á
ˉ
100
How will oxide growth rate be affected by boron atoms?
Highly doped substrates oxidize more rapidly than do lightly doped wafers,At lower
temperatures,and for thinner oxides,the enhancement can be three to four times that for
undoped Si.
3
B
A
=
B
A
ê
á
ˉ
i
1+ 2.62 ￥10
3
exp -
1.1eV
kT
ê
á
ˉ
C
V
T
C
V
i
T
-1
ê
á
á
ˉ
è
í
í
í
˙
˙
˙
The boron dopant concentration is implied by the usage of the term,heavily-doped”.
Heavily-doped means degenerately-doped,which is doping greater than 10
19
cm
-3
(page
22 in Plummer).
This equation and definition are provided for reference,but to solve this problem,we're
not expecting you to use the equation,Instead,using a physical understanding of the
oxidation process,calculate the oxide thickness for typical silicon material and then
acknowledge that this is a lower bound,and then qualitatively explain how the
polycrystalline nature and doping affect this bound,One could also look at the lecture
notes that show how oxidation thickness is affected by doping and also how B/A and B
are affected.
