Optimization of the Czochralski method for InSb crystal ... - CUMC
Optimization of the Czochralski method for InSb crystal ... - CUMC
Optimization of the Czochralski method for InSb crystal ... - CUMC
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Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong><br />
<strong>crystal</strong> growth<br />
<strong>CUMC</strong> 2008<br />
July 9 to 12<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
University <strong>of</strong> Ontario Institute <strong>of</strong> Technology<br />
July 9, 2008<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
Outline<br />
1 Background<br />
Commercial uses <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Benefits and detriments <strong>of</strong> using <strong>InSb</strong><br />
2 Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
The model<br />
Future work<br />
3 Summary<br />
In conclusion<br />
4 References<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
Commercial uses <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Benefits and detriments <strong>of</strong> using <strong>InSb</strong><br />
Commercial uses include<br />
Thermal radiation detection<br />
Military gear<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
Commercial uses <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Benefits and detriments <strong>of</strong> using <strong>InSb</strong><br />
Commercial uses include<br />
Thermal radiation detection<br />
Military gear<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
Commercial uses <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Benefits and detriments <strong>of</strong> using <strong>InSb</strong><br />
Commercial uses include<br />
Thermal radiation detection<br />
Military gear<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
Benefits and detriments<br />
Commercial uses <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Benefits and detriments <strong>of</strong> using <strong>InSb</strong><br />
Benefits <strong>of</strong> using <strong>InSb</strong> <strong>crystal</strong><br />
Easy passage <strong>of</strong> electrons<br />
Detriments <strong>of</strong> using <strong>InSb</strong> <strong>crystal</strong><br />
Difficult to grow with Cz technique<br />
Loose structure causes stress marks in large <strong>crystal</strong>s<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
Benefits and detriments<br />
Commercial uses <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Benefits and detriments <strong>of</strong> using <strong>InSb</strong><br />
Benefits <strong>of</strong> using <strong>InSb</strong> <strong>crystal</strong><br />
Easy passage <strong>of</strong> electrons<br />
Detriments <strong>of</strong> using <strong>InSb</strong> <strong>crystal</strong><br />
Difficult to grow with Cz technique<br />
Loose structure causes stress marks in large <strong>crystal</strong>s<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
Benefits and detriments<br />
Commercial uses <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Benefits and detriments <strong>of</strong> using <strong>InSb</strong><br />
Benefits <strong>of</strong> using <strong>InSb</strong> <strong>crystal</strong><br />
Easy passage <strong>of</strong> electrons<br />
Detriments <strong>of</strong> using <strong>InSb</strong> <strong>crystal</strong><br />
Difficult to grow with Cz technique<br />
Loose structure causes stress marks in large <strong>crystal</strong>s<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
Benefits and detriments<br />
Commercial uses <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Benefits and detriments <strong>of</strong> using <strong>InSb</strong><br />
Benefits <strong>of</strong> using <strong>InSb</strong> <strong>crystal</strong><br />
Easy passage <strong>of</strong> electrons<br />
Detriments <strong>of</strong> using <strong>InSb</strong> <strong>crystal</strong><br />
Difficult to grow with Cz technique<br />
Loose structure causes stress marks in large <strong>crystal</strong>s<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
Benefits and detriments<br />
Commercial uses <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Benefits and detriments <strong>of</strong> using <strong>InSb</strong><br />
Benefits <strong>of</strong> using <strong>InSb</strong> <strong>crystal</strong><br />
Easy passage <strong>of</strong> electrons<br />
Detriments <strong>of</strong> using <strong>InSb</strong> <strong>crystal</strong><br />
Difficult to grow with Cz technique<br />
Loose structure causes stress marks in large <strong>crystal</strong>s<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
Benefits and detriments<br />
Commercial uses <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Benefits and detriments <strong>of</strong> using <strong>InSb</strong><br />
Benefits <strong>of</strong> using <strong>InSb</strong> <strong>crystal</strong><br />
Easy passage <strong>of</strong> electrons<br />
Detriments <strong>of</strong> using <strong>InSb</strong> <strong>crystal</strong><br />
Difficult to grow with Cz technique<br />
Loose structure causes stress marks in large <strong>crystal</strong>s<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
<strong>Czochralski</strong> growth <strong>of</strong> ruby<br />
Commercial uses <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Benefits and detriments <strong>of</strong> using <strong>InSb</strong><br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
<strong>InSb</strong> <strong>crystal</strong><br />
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
Commercial uses <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Benefits and detriments <strong>of</strong> using <strong>InSb</strong><br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Basic assumptions<br />
1 Crystal is axis-symmetric<br />
2 Heat exchange is constant and uni<strong>for</strong>m<br />
3 Mean <strong>crystal</strong> radius is small compared to its length<br />
4 Thermal stress is elastic<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Basic assumptions<br />
1 Crystal is axis-symmetric<br />
2 Heat exchange is constant and uni<strong>for</strong>m<br />
3 Mean <strong>crystal</strong> radius is small compared to its length<br />
4 Thermal stress is elastic<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Basic assumptions<br />
1 Crystal is axis-symmetric<br />
2 Heat exchange is constant and uni<strong>for</strong>m<br />
3 Mean <strong>crystal</strong> radius is small compared to its length<br />
4 Thermal stress is elastic<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Basic assumptions<br />
1 Crystal is axis-symmetric<br />
2 Heat exchange is constant and uni<strong>for</strong>m<br />
3 Mean <strong>crystal</strong> radius is small compared to its length<br />
4 Thermal stress is elastic<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Basic assumptions<br />
1 Crystal is axis-symmetric<br />
2 Heat exchange is constant and uni<strong>for</strong>m<br />
3 Mean <strong>crystal</strong> radius is small compared to its length<br />
4 Thermal stress is elastic<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Variables to consider<br />
Orientation<br />
Shape<br />
Length<br />
Radius<br />
Growth rate<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Variables to consider<br />
Orientation<br />
Shape<br />
Length<br />
Radius<br />
Growth rate<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Variables to consider<br />
Orientation<br />
Shape<br />
Length<br />
Radius<br />
Growth rate<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Variables to consider<br />
Orientation<br />
Shape<br />
Length<br />
Radius<br />
Growth rate<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Variables to consider<br />
Orientation<br />
Shape<br />
Length<br />
Radius<br />
Growth rate<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Variables to consider<br />
Orientation<br />
Shape<br />
Length<br />
Radius<br />
Growth rate<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Orientation<br />
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Alignment affects stress points in <strong>crystal</strong><br />
[001] ε1<br />
0.03<br />
0.045<br />
0.04<br />
0.035<br />
0.03<br />
[001] ε2<br />
0.18<br />
0.16<br />
0.14<br />
0.12<br />
0<br />
0.025<br />
0.02<br />
0.1<br />
0.08<br />
0.015<br />
0.06<br />
0.01<br />
0.04<br />
0.005<br />
0.02<br />
-0.03<br />
0<br />
0<br />
0.03<br />
[211] ε1<br />
0.045<br />
0.18<br />
0.04<br />
0.16<br />
0.035<br />
0.14<br />
6<br />
4<br />
0<br />
0.03<br />
0.025<br />
0.02<br />
[211] ε 2<br />
0<br />
0.12<br />
0.1<br />
0.08<br />
0.015<br />
0.06<br />
2<br />
0.01<br />
0.04<br />
0.005<br />
0.02<br />
0<br />
-0.03<br />
0<br />
[111] ε 1<br />
0<br />
[111] ε 2<br />
0<br />
-2<br />
0.03<br />
0.045<br />
0.18<br />
0.04<br />
0.16<br />
-4<br />
0.035<br />
0.14<br />
-6<br />
6<br />
4<br />
2<br />
0<br />
-2<br />
-4<br />
-6<br />
-6<br />
-4<br />
-2<br />
0<br />
2<br />
4<br />
6<br />
0<br />
-0.03<br />
0.03<br />
0.025<br />
0.02<br />
0.015<br />
0.01<br />
0.005<br />
0.12<br />
0.1<br />
0.08<br />
0.06<br />
0.04<br />
0.02<br />
“Best” orientation [¯211]<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Orientation<br />
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Alignment affects stress points in <strong>crystal</strong><br />
[001] ε1<br />
0.03<br />
0.045<br />
0.04<br />
0.035<br />
0.03<br />
[001] ε2<br />
0.18<br />
0.16<br />
0.14<br />
0.12<br />
0<br />
0.025<br />
0.02<br />
0.1<br />
0.08<br />
0.015<br />
0.06<br />
0.01<br />
0.04<br />
0.005<br />
0.02<br />
-0.03<br />
0<br />
0<br />
0.03<br />
[211] ε1<br />
0.045<br />
0.18<br />
0.04<br />
0.16<br />
0.035<br />
0.14<br />
6<br />
4<br />
0<br />
0.03<br />
0.025<br />
0.02<br />
[211] ε 2<br />
0<br />
0.12<br />
0.1<br />
0.08<br />
0.015<br />
0.06<br />
2<br />
0.01<br />
0.04<br />
0.005<br />
0.02<br />
0<br />
-0.03<br />
0<br />
[111] ε 1<br />
0<br />
[111] ε 2<br />
0<br />
-2<br />
0.03<br />
0.045<br />
0.18<br />
0.04<br />
0.16<br />
-4<br />
0.035<br />
0.14<br />
-6<br />
6<br />
4<br />
2<br />
0<br />
-2<br />
-4<br />
-6<br />
-6<br />
-4<br />
-2<br />
0<br />
2<br />
4<br />
6<br />
0<br />
-0.03<br />
0.03<br />
0.025<br />
0.02<br />
0.015<br />
0.01<br />
0.005<br />
0.12<br />
0.1<br />
0.08<br />
0.06<br />
0.04<br />
0.02<br />
“Best” orientation [¯211]<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Shape<br />
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Affects stresses on edges <strong>of</strong> <strong>crystal</strong><br />
Shapes compared:<br />
Cylinder<br />
Cone<br />
‘Best’ shape:<br />
Cone<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Shape<br />
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Affects stresses on edges <strong>of</strong> <strong>crystal</strong><br />
Shapes compared:<br />
Cylinder<br />
Cone<br />
‘Best’ shape:<br />
Cone<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Shape<br />
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Affects stresses on edges <strong>of</strong> <strong>crystal</strong><br />
Shapes compared:<br />
Cylinder<br />
Cone<br />
‘Best’ shape:<br />
Cone<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Shape<br />
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Affects stresses on edges <strong>of</strong> <strong>crystal</strong><br />
Shapes compared:<br />
Cylinder<br />
Cone<br />
‘Best’ shape:<br />
Cone<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Shape<br />
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Affects stresses on edges <strong>of</strong> <strong>crystal</strong><br />
Shapes compared:<br />
Cylinder<br />
Cone<br />
‘Best’ shape:<br />
Cone<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Shape<br />
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Affects stresses on edges <strong>of</strong> <strong>crystal</strong><br />
Shapes compared:<br />
Cylinder<br />
Cone<br />
‘Best’ shape:<br />
Cone<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Shape<br />
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Affects stresses on edges <strong>of</strong> <strong>crystal</strong><br />
Shapes compared:<br />
Cylinder<br />
Cone<br />
‘Best’ shape:<br />
Cone<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Cylindrical<br />
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
1<br />
unsteady<br />
pseudo-steady<br />
1<br />
2<br />
unsteady<br />
pseudo-steady<br />
2<br />
1<br />
0.9<br />
ε = ε 1 ε = ε 1 ε = ε 2 ε = ε 2<br />
0.9<br />
1.8<br />
1.8<br />
0.9<br />
0.8<br />
0.8<br />
1.6<br />
1.6<br />
0.8<br />
0.7<br />
0.7<br />
1.4<br />
1.4<br />
0.7<br />
0.6<br />
0.6<br />
1.2<br />
1.2<br />
0.6<br />
z<br />
0.5<br />
z<br />
0.5<br />
z<br />
1<br />
z<br />
1<br />
0.5<br />
0.4<br />
0.4<br />
0.8<br />
0.8<br />
0.4<br />
0.3<br />
0.3<br />
0.6<br />
0.6<br />
0.3<br />
0.2<br />
0.2<br />
0.4<br />
0.4<br />
0.2<br />
0.1<br />
0.1<br />
0.2<br />
0.2<br />
0.1<br />
0<br />
0 1 2<br />
r<br />
0<br />
0 1 2<br />
r<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
0<br />
0 1 2<br />
r<br />
0<br />
0 1 2<br />
r<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth<br />
0
Conical<br />
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
1<br />
unsteady<br />
pseudo-steady<br />
1<br />
2<br />
unsteady<br />
pseudo-steady<br />
2<br />
1<br />
0.9<br />
ε = ε 1 ε = ε 1 ε = ε 2 ε = ε 2<br />
0.9<br />
1.8<br />
1.8<br />
0.9<br />
0.8<br />
0.8<br />
1.6<br />
1.6<br />
0.8<br />
0.7<br />
0.7<br />
1.4<br />
1.4<br />
0.7<br />
0.6<br />
0.6<br />
1.2<br />
1.2<br />
0.6<br />
z<br />
0.5<br />
z<br />
0.5<br />
z<br />
1<br />
z<br />
1<br />
0.5<br />
0.4<br />
0.4<br />
0.8<br />
0.8<br />
0.4<br />
0.3<br />
0.3<br />
0.6<br />
0.6<br />
0.3<br />
0.2<br />
0.2<br />
0.4<br />
0.4<br />
0.2<br />
0.1<br />
0.1<br />
0.2<br />
0.2<br />
0.1<br />
0<br />
0 1 2<br />
r<br />
0<br />
0 1 2<br />
r<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
0<br />
0 1 2<br />
r<br />
0<br />
0 1 2<br />
r<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth<br />
0
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Stress on cylindrical and conical <strong>crystal</strong>s<br />
1<br />
0.9<br />
0.8<br />
1<br />
2<br />
2<br />
ε = ε 1<br />
ε = ε 1 ε = ε 2 ε = ε 2<br />
0.9<br />
0.16 1.8<br />
1.8<br />
0.8<br />
0.14 1.6<br />
1.6<br />
0.7<br />
0.6<br />
0.7<br />
0.7<br />
0.12<br />
1.4<br />
1.4<br />
0.5<br />
0.6<br />
0.6<br />
0.1<br />
1.2<br />
1.2<br />
0.4<br />
z<br />
0.5<br />
0.4<br />
z<br />
0.5<br />
0.4<br />
1<br />
0.08<br />
0.8<br />
z<br />
z<br />
1<br />
0.8<br />
0.3<br />
0.3<br />
0.2<br />
0.3<br />
0.2<br />
0.06<br />
0.04<br />
0.6<br />
0.4<br />
0.6<br />
0.4<br />
0.2<br />
0.1<br />
0.1<br />
0.02<br />
0.2<br />
0.2<br />
0.1<br />
0<br />
0 1 2<br />
r<br />
0<br />
0 1 2<br />
r<br />
0<br />
0<br />
0 1 2<br />
r<br />
0<br />
0 1 2<br />
r<br />
0<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Length and radius<br />
Larger <strong>crystal</strong>s have more stress<br />
Greater temperature change with respect to radius<br />
Larger <strong>crystal</strong>s needed <strong>for</strong> commercial purposes<br />
Fewer boundary effects<br />
Clearer reception<br />
Larger receptors<br />
Current largest clean <strong>crystal</strong> grown commercially: 10 cm<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Length and radius<br />
Larger <strong>crystal</strong>s have more stress<br />
Greater temperature change with respect to radius<br />
Larger <strong>crystal</strong>s needed <strong>for</strong> commercial purposes<br />
Fewer boundary effects<br />
Clearer reception<br />
Larger receptors<br />
Current largest clean <strong>crystal</strong> grown commercially: 10 cm<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Length and radius<br />
Larger <strong>crystal</strong>s have more stress<br />
Greater temperature change with respect to radius<br />
Larger <strong>crystal</strong>s needed <strong>for</strong> commercial purposes<br />
Fewer boundary effects<br />
Clearer reception<br />
Larger receptors<br />
Current largest clean <strong>crystal</strong> grown commercially: 10 cm<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Length and radius<br />
Larger <strong>crystal</strong>s have more stress<br />
Greater temperature change with respect to radius<br />
Larger <strong>crystal</strong>s needed <strong>for</strong> commercial purposes<br />
Fewer boundary effects<br />
Clearer reception<br />
Larger receptors<br />
Current largest clean <strong>crystal</strong> grown commercially: 10 cm<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Length and radius<br />
Larger <strong>crystal</strong>s have more stress<br />
Greater temperature change with respect to radius<br />
Larger <strong>crystal</strong>s needed <strong>for</strong> commercial purposes<br />
Fewer boundary effects<br />
Clearer reception<br />
Larger receptors<br />
Current largest clean <strong>crystal</strong> grown commercially: 10 cm<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Length and radius<br />
Larger <strong>crystal</strong>s have more stress<br />
Greater temperature change with respect to radius<br />
Larger <strong>crystal</strong>s needed <strong>for</strong> commercial purposes<br />
Fewer boundary effects<br />
Clearer reception<br />
Larger receptors<br />
Current largest clean <strong>crystal</strong> grown commercially: 10 cm<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Length and radius<br />
Larger <strong>crystal</strong>s have more stress<br />
Greater temperature change with respect to radius<br />
Larger <strong>crystal</strong>s needed <strong>for</strong> commercial purposes<br />
Fewer boundary effects<br />
Clearer reception<br />
Larger receptors<br />
Current largest clean <strong>crystal</strong> grown commercially: 10 cm<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Length and radius<br />
Larger <strong>crystal</strong>s have more stress<br />
Greater temperature change with respect to radius<br />
Larger <strong>crystal</strong>s needed <strong>for</strong> commercial purposes<br />
Fewer boundary effects<br />
Clearer reception<br />
Larger receptors<br />
Current largest clean <strong>crystal</strong> grown commercially: 10 cm<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Growth rate<br />
Slower growth rate creates cleaner <strong>crystal</strong>s<br />
Impractical<br />
Necessary to grow faster <strong>for</strong> commercial purposes<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Growth rate<br />
Slower growth rate creates cleaner <strong>crystal</strong>s<br />
Impractical<br />
Necessary to grow faster <strong>for</strong> commercial purposes<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Growth rate<br />
Slower growth rate creates cleaner <strong>crystal</strong>s<br />
Impractical<br />
Necessary to grow faster <strong>for</strong> commercial purposes<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Growth rate<br />
Slower growth rate creates cleaner <strong>crystal</strong>s<br />
Impractical<br />
Necessary to grow faster <strong>for</strong> commercial purposes<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Strain equations<br />
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Strain is relative displacement<br />
Example: If a L = 1m long bar stretches ∆L = 1cm <strong>the</strong>n <strong>the</strong><br />
strain is ∆L/L = 0.01.<br />
Derived from displacement vector u : R 3 ↦→ R 3<br />
Dependent on <strong>the</strong> <strong>the</strong>rmal perturbations<br />
One possible representation<br />
⎛<br />
⎞<br />
e rr e rφ e rz<br />
e = ⎝e φr e φφ e φz<br />
⎠ , e T = e.<br />
e zr e zφ e zz<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Strain equations<br />
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Strain is relative displacement<br />
Example: If a L = 1m long bar stretches ∆L = 1cm <strong>the</strong>n <strong>the</strong><br />
strain is ∆L/L = 0.01.<br />
Derived from displacement vector u : R 3 ↦→ R 3<br />
Dependent on <strong>the</strong> <strong>the</strong>rmal perturbations<br />
One possible representation<br />
⎛<br />
⎞<br />
e rr e rφ e rz<br />
e = ⎝e φr e φφ e φz<br />
⎠ , e T = e.<br />
e zr e zφ e zz<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Strain equations<br />
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Strain is relative displacement<br />
Example: If a L = 1m long bar stretches ∆L = 1cm <strong>the</strong>n <strong>the</strong><br />
strain is ∆L/L = 0.01.<br />
Derived from displacement vector u : R 3 ↦→ R 3<br />
Dependent on <strong>the</strong> <strong>the</strong>rmal perturbations<br />
One possible representation<br />
⎛<br />
⎞<br />
e rr e rφ e rz<br />
e = ⎝e φr e φφ e φz<br />
⎠ , e T = e.<br />
e zr e zφ e zz<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Strain equations<br />
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Strain is relative displacement<br />
Example: If a L = 1m long bar stretches ∆L = 1cm <strong>the</strong>n <strong>the</strong><br />
strain is ∆L/L = 0.01.<br />
Derived from displacement vector u : R 3 ↦→ R 3<br />
Dependent on <strong>the</strong> <strong>the</strong>rmal perturbations<br />
One possible representation<br />
⎛<br />
⎞<br />
e rr e rφ e rz<br />
e = ⎝e φr e φφ e φz<br />
⎠ , e T = e.<br />
e zr e zφ e zz<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Strain equations<br />
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Strain is relative displacement<br />
Example: If a L = 1m long bar stretches ∆L = 1cm <strong>the</strong>n <strong>the</strong><br />
strain is ∆L/L = 0.01.<br />
Derived from displacement vector u : R 3 ↦→ R 3<br />
Dependent on <strong>the</strong> <strong>the</strong>rmal perturbations<br />
One possible representation<br />
⎛<br />
⎞<br />
e rr e rφ e rz<br />
e = ⎝e φr e φφ e φz<br />
⎠ , e T = e.<br />
e zr e zφ e zz<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Strain equations<br />
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Strain is relative displacement<br />
Example: If a L = 1m long bar stretches ∆L = 1cm <strong>the</strong>n <strong>the</strong><br />
strain is ∆L/L = 0.01.<br />
Derived from displacement vector u : R 3 ↦→ R 3<br />
Dependent on <strong>the</strong> <strong>the</strong>rmal perturbations<br />
One possible representation<br />
⎛<br />
⎞<br />
e rr e rφ e rz<br />
e = ⎝e φr e φφ e φz<br />
⎠ , e T = e.<br />
e zr e zφ e zz<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
In our case, with displacement vector u = 〈u(r), 0, 0〉 in<br />
cylindrical coordinates,<br />
e rr = ∂u<br />
∂r → u r ,<br />
e zz = ∂w<br />
∂z → 0,<br />
2e rz = ∂u<br />
∂z + ∂w<br />
∂r<br />
→ 0,<br />
e φφ = 1 ∂v<br />
r ∂φ + u r → u r ,<br />
2e φz = 1 ∂w<br />
r ∂φ + ∂v<br />
∂z → 0,<br />
2e rφ = ∂v<br />
∂r − v r + 1 ∂u<br />
r ∂φ → 0.<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Stress equations<br />
Pressures (stress) that give rise to relative displacements<br />
(strain)<br />
Dependent on strain equations<br />
General matrix equation (isotropic material)<br />
σ ij =<br />
where<br />
E<br />
(1 + ν) e Eν<br />
ij +<br />
(1 + ν)(1 − 2ν) e llδ ij − α 0(T − T g )E<br />
δ ij ,<br />
(1 − 2ν)<br />
or, in our case,<br />
e ll = e rr + e φφ + e zz , δ ij =<br />
e ll = u r + u r<br />
{<br />
1 i = j<br />
0 i ≠ j<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Stress equations<br />
Pressures (stress) that give rise to relative displacements<br />
(strain)<br />
Dependent on strain equations<br />
General matrix equation (isotropic material)<br />
σ ij =<br />
where<br />
E<br />
(1 + ν) e Eν<br />
ij +<br />
(1 + ν)(1 − 2ν) e llδ ij − α 0(T − T g )E<br />
δ ij ,<br />
(1 − 2ν)<br />
or, in our case,<br />
e ll = e rr + e φφ + e zz , δ ij =<br />
e ll = u r + u r<br />
{<br />
1 i = j<br />
0 i ≠ j<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Stress equations<br />
Pressures (stress) that give rise to relative displacements<br />
(strain)<br />
Dependent on strain equations<br />
General matrix equation (isotropic material)<br />
σ ij =<br />
where<br />
E<br />
(1 + ν) e Eν<br />
ij +<br />
(1 + ν)(1 − 2ν) e llδ ij − α 0(T − T g )E<br />
δ ij ,<br />
(1 − 2ν)<br />
or, in our case,<br />
e ll = e rr + e φφ + e zz , δ ij =<br />
e ll = u r + u r<br />
{<br />
1 i = j<br />
0 i ≠ j<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Stress equations<br />
Pressures (stress) that give rise to relative displacements<br />
(strain)<br />
Dependent on strain equations<br />
General matrix equation (isotropic material)<br />
σ ij =<br />
where<br />
E<br />
(1 + ν) e Eν<br />
ij +<br />
(1 + ν)(1 − 2ν) e llδ ij − α 0(T − T g )E<br />
δ ij ,<br />
(1 − 2ν)<br />
or, in our case,<br />
e ll = e rr + e φφ + e zz , δ ij =<br />
e ll = u r + u r<br />
{<br />
1 i = j<br />
0 i ≠ j<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Stress equations<br />
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Stress-strain (constitutive) relationship depends on material<br />
More convenient representation <strong>for</strong> isotropic material<br />
where<br />
σ = Ce − α 0 (T − T g )(C 11 + 2C 12 )I,<br />
⎛<br />
⎞<br />
C 11 C 12 C 12 0 0 0<br />
C 12 C 11 C 12 0 0 0<br />
C =<br />
C 12 C 12 C 11 0 0 0<br />
⎜ 0 0 0 C 44 0 0<br />
,<br />
⎟<br />
⎝ 0 0 0 0 C 44 0 ⎠<br />
0 0 0 0 0 C 44<br />
e T = (e rr , e φφ , e zz , 2e φz , 2e rz , 2e rφ ),<br />
σ T = (σ rr , σ φφ , σ zz , σ φz , σ rz , σ rφ ).<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Stress equations<br />
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Stress-strain (constitutive) relationship depends on material<br />
More convenient representation <strong>for</strong> isotropic material<br />
where<br />
σ = Ce − α 0 (T − T g )(C 11 + 2C 12 )I,<br />
⎛<br />
⎞<br />
C 11 C 12 C 12 0 0 0<br />
C 12 C 11 C 12 0 0 0<br />
C =<br />
C 12 C 12 C 11 0 0 0<br />
⎜ 0 0 0 C 44 0 0<br />
,<br />
⎟<br />
⎝ 0 0 0 0 C 44 0 ⎠<br />
0 0 0 0 0 C 44<br />
e T = (e rr , e φφ , e zz , 2e φz , 2e rz , 2e rφ ),<br />
σ T = (σ rr , σ φφ , σ zz , σ φz , σ rz , σ rφ ).<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Stress equations<br />
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
In our case (isotropic: 2C 44 − C 11 + C 12 )<br />
which gives us<br />
σ ij =<br />
as be<strong>for</strong>e.<br />
C 11 =<br />
C 12 =<br />
C 44 =<br />
(1 − ν)E<br />
(1 + ν)(1 − 2ν) ,<br />
νE<br />
(1 + ν)(1 − 2ν) ,<br />
E<br />
2(1 + ν) ,<br />
E<br />
(1 + ν) e Eν<br />
ij +<br />
(1 + ν)(1 − 2ν) e llδ ij − α 0(T − T g )E<br />
δ ij<br />
(1 − 2ν)<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
von Mises stress<br />
Used to find <strong>the</strong> maximal stress<br />
von Mises stress equation<br />
For us<br />
σ VM = (σ φφ − σ zz ) 2 + (σ rr − σ zz ) 2 + (σ rr − σ φφ ) 2<br />
2σ VM = 1 4 ε2 Θ 12<br />
1 r 4 + 1 16 ε2 Θ 12<br />
1 (3r 2 − R 2 ) 2 + 1<br />
16 ε2 Θ 12<br />
1 (r 2 − R 2 ) 2<br />
or<br />
σ VM = 1 4 ε ∣ ∣Θ 1 (<br />
∣ r<br />
) 2 ( r<br />
) ) 4<br />
1 R<br />
(1 2 − 4 + 7 .<br />
R R<br />
with Θ 1 1 ≃ ∂T /∂r<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
von Mises stress<br />
Used to find <strong>the</strong> maximal stress<br />
von Mises stress equation<br />
For us<br />
σ VM = (σ φφ − σ zz ) 2 + (σ rr − σ zz ) 2 + (σ rr − σ φφ ) 2<br />
2σ VM = 1 4 ε2 Θ 12<br />
1 r 4 + 1 16 ε2 Θ 12<br />
1 (3r 2 − R 2 ) 2 + 1<br />
16 ε2 Θ 12<br />
1 (r 2 − R 2 ) 2<br />
or<br />
σ VM = 1 4 ε ∣ ∣Θ 1 (<br />
∣ r<br />
) 2 ( r<br />
) ) 4<br />
1 R<br />
(1 2 − 4 + 7 .<br />
R R<br />
with Θ 1 1 ≃ ∂T /∂r<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
von Mises stress<br />
Used to find <strong>the</strong> maximal stress<br />
von Mises stress equation<br />
For us<br />
σ VM = (σ φφ − σ zz ) 2 + (σ rr − σ zz ) 2 + (σ rr − σ φφ ) 2<br />
2σ VM = 1 4 ε2 Θ 12<br />
1 r 4 + 1 16 ε2 Θ 12<br />
1 (3r 2 − R 2 ) 2 + 1<br />
16 ε2 Θ 12<br />
1 (r 2 − R 2 ) 2<br />
or<br />
σ VM = 1 4 ε ∣ ∣Θ 1 (<br />
∣ r<br />
) 2 ( r<br />
) ) 4<br />
1 R<br />
(1 2 − 4 + 7 .<br />
R R<br />
with Θ 1 1 ≃ ∂T /∂r<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
von Mises stress<br />
Used to find <strong>the</strong> maximal stress<br />
von Mises stress equation<br />
For us<br />
σ VM = (σ φφ − σ zz ) 2 + (σ rr − σ zz ) 2 + (σ rr − σ φφ ) 2<br />
2σ VM = 1 4 ε2 Θ 12<br />
1 r 4 + 1 16 ε2 Θ 12<br />
1 (3r 2 − R 2 ) 2 + 1<br />
16 ε2 Θ 12<br />
1 (r 2 − R 2 ) 2<br />
or<br />
σ VM = 1 4 ε ∣ ∣Θ 1 (<br />
∣ r<br />
) 2 ( r<br />
) ) 4<br />
1 R<br />
(1 2 − 4 + 7 .<br />
R R<br />
with Θ 1 1 ≃ ∂T /∂r<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Fur<strong>the</strong>r studies include<br />
Minimizing maximum von Mises stress<br />
Dynamically investigating optimal shape<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Fur<strong>the</strong>r studies include<br />
Minimizing maximum von Mises stress<br />
Dynamically investigating optimal shape<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
The model<br />
Future work<br />
Fur<strong>the</strong>r studies include<br />
Minimizing maximum von Mises stress<br />
Dynamically investigating optimal shape<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
In conclusion<br />
To sum up<br />
We have<br />
found quick <strong>method</strong> <strong>of</strong> predicting stress <strong>for</strong> any given pr<strong>of</strong>ile<br />
removed axis symmetry condition (facets)<br />
removed isotropy condition (splitting C = C 0 − C a )<br />
We are working on<br />
minimizing von Mises stress<br />
finding optimal growth rate and shape up to second order<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
In conclusion<br />
To sum up<br />
We have<br />
found quick <strong>method</strong> <strong>of</strong> predicting stress <strong>for</strong> any given pr<strong>of</strong>ile<br />
removed axis symmetry condition (facets)<br />
removed isotropy condition (splitting C = C 0 − C a )<br />
We are working on<br />
minimizing von Mises stress<br />
finding optimal growth rate and shape up to second order<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
In conclusion<br />
To sum up<br />
We have<br />
found quick <strong>method</strong> <strong>of</strong> predicting stress <strong>for</strong> any given pr<strong>of</strong>ile<br />
removed axis symmetry condition (facets)<br />
removed isotropy condition (splitting C = C 0 − C a )<br />
We are working on<br />
minimizing von Mises stress<br />
finding optimal growth rate and shape up to second order<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
In conclusion<br />
To sum up<br />
We have<br />
found quick <strong>method</strong> <strong>of</strong> predicting stress <strong>for</strong> any given pr<strong>of</strong>ile<br />
removed axis symmetry condition (facets)<br />
removed isotropy condition (splitting C = C 0 − C a )<br />
We are working on<br />
minimizing von Mises stress<br />
finding optimal growth rate and shape up to second order<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
In conclusion<br />
To sum up<br />
We have<br />
found quick <strong>method</strong> <strong>of</strong> predicting stress <strong>for</strong> any given pr<strong>of</strong>ile<br />
removed axis symmetry condition (facets)<br />
removed isotropy condition (splitting C = C 0 − C a )<br />
We are working on<br />
minimizing von Mises stress<br />
finding optimal growth rate and shape up to second order<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
In conclusion<br />
To sum up<br />
We have<br />
found quick <strong>method</strong> <strong>of</strong> predicting stress <strong>for</strong> any given pr<strong>of</strong>ile<br />
removed axis symmetry condition (facets)<br />
removed isotropy condition (splitting C = C 0 − C a )<br />
We are working on<br />
minimizing von Mises stress<br />
finding optimal growth rate and shape up to second order<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
In conclusion<br />
To sum up<br />
We have<br />
found quick <strong>method</strong> <strong>of</strong> predicting stress <strong>for</strong> any given pr<strong>of</strong>ile<br />
removed axis symmetry condition (facets)<br />
removed isotropy condition (splitting C = C 0 − C a )<br />
We are working on<br />
minimizing von Mises stress<br />
finding optimal growth rate and shape up to second order<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
In conclusion<br />
To sum up<br />
We have<br />
found quick <strong>method</strong> <strong>of</strong> predicting stress <strong>for</strong> any given pr<strong>of</strong>ile<br />
removed axis symmetry condition (facets)<br />
removed isotropy condition (splitting C = C 0 − C a )<br />
We are working on<br />
minimizing von Mises stress<br />
finding optimal growth rate and shape up to second order<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
In conclusion<br />
To sum up<br />
We have<br />
found quick <strong>method</strong> <strong>of</strong> predicting stress <strong>for</strong> any given pr<strong>of</strong>ile<br />
removed axis symmetry condition (facets)<br />
removed isotropy condition (splitting C = C 0 − C a )<br />
We are working on<br />
minimizing von Mises stress<br />
finding optimal growth rate and shape up to second order<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
Bohun, C.S. Stress analysis <strong>of</strong> Cz grown <strong>InSb</strong>, not published.<br />
Bohun, C.S., Frigaard, I., Huang, H., and Liang, S. (2006) A<br />
semianalytical <strong>the</strong>rmal stress model <strong>for</strong> <strong>the</strong> <strong>Czochralski</strong> growth<br />
<strong>of</strong> type III-V compounds, SIAM J Appl. Math, vol. 66, no. 5,<br />
pp. 1533-1562.<br />
Huang, H. and Liang, S. (2007) Thermal-stress reduction <strong>for</strong> a<br />
<strong>Czochralski</strong> grown single <strong>crystal</strong>, Springer J Eng Math, vol. 59,<br />
pp. 1-23.<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
Background<br />
Analysis <strong>of</strong> <strong>InSb</strong> <strong>crystal</strong><br />
Summary<br />
References<br />
Thank you<br />
Devon Armstrong, Faculty <strong>of</strong> Science<br />
<strong>Optimization</strong> <strong>of</strong> <strong>the</strong> <strong>Czochralski</strong> <strong>method</strong> <strong>for</strong> <strong>InSb</strong> <strong>crystal</strong> growth
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