Bermel ECE 305 S18
ECE-305: Spring 2018
Material Properties
Professor Peter BermelElectrical and Computer Engineering
Purdue University, West Lafayette, IN [email protected]
Pierret, Semiconductor Device Fundamentals (SDF)Chapter 1 (pp. 3-19)Chapter 2 (pp. 22-32)
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Bermel ECE 305 S18 2
outline
1. Graphene
2. Silicon
3. Miller indices
4. Quantization of energy levels
5. Energy bands
6. Electrons and holes
7. Intrinsic carriers
8. Doping
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semiconductors
3http://en.wikipedia.org/wiki/Periodic_table
column4
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4
carbon energy levels
1s2
2s2
2p64 valence electrons8 valence states
“core level”
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energ
y
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Graphene: 2011 Nobel Prize in Physics
5
Graphene is a one-atom-thick planar crystalline carbon sheet with a triangular lattice with 2 atoms per unit cell.
source: CNTBands 2.7.2
https://nanohub.org/resources/1838
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triangular lattice + 2 atom basis
6Bermel ECE 305 S18
a
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a
Bermel ECE 305 S18 7
outline
1. Graphene
2. Silicon
3. Miller indices
4. Quantization of energy levels
5. Energy bands
6. Electrons and holes
7. Intrinsic carriers
8. Doping
1/11/2018
✔
8
silicon energy levels
1s2
2s2
2p6
3s2
3p2
4s0
4 valence electrons8 valence states
“core levels”
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energ
y
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“cartoon” Si crystal
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What would you get from combining carbon in the same way?
3D crystal structure
10http://en.wikipedia.org/wiki/Bravais_lattice
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silicon in diamond lattice
https://nanohub.org/tools/crystal_viewer
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The diamond lattice
https://nanohub.org/tools/crystal_viewer
Atoms per unit cell
8 times 1/8 + 6 times ½ + 4
8 atoms per unit cell
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Silicon: nearest neighbor (NN) spacing
https://nanohub.org/tools/crystal_viewer
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Si atoms in a solid
18
1) In a Si crystal, each atom occupies, a specific location in a crystal lattice.
2) Polycrystalline Si consider of many crystalline “grains” with different orientations.
3) In amorphous Si, the atoms are more or less randomly distributed throughout the solid.
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semiconductors
19http://en.wikipedia.org/wiki/Periodic_table
column4
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semiconductors
20http://en.wikipedia.org/wiki/Periodic_table
Col.3
Col.5
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Bermel ECE 305 S18 21
outline
1. Graphene
2. Silicon
3. Miller indices
4. Quantization of energy levels
5. Energy bands
6. Electrons and holes
7. Intrinsic carriers
8. Doping
1/11/2018
✔✔
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Miller index prescription for describing planes
x
y
z
2a
a
2a
x, y, and z-axis intercepts:
2a, 1a, 2a2, 1, 2
invert:
½. 1, ½
Rationalize:
1, 2, 1
(1, 2, 1) plane
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question
Where does this prescription come from?
Answer: If we remember the equation for a plane, we can figure it out.
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where it comes from
x
y
z
2a
a
2a
(1, 2, 1) plane
equation of a plane:
x
xint
+y
yint
+z
zint
= 1
describe with the numbers:
1
xint
,1
yint
,1
zint
equivalent to:
1
xint
a,
1
yint
a,
1
zint
a
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http://www.doitpoms.ac.uk/tlplib/miller_indices/lattice_draw.php
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prescription for describing directions
x
y
z
3a
2a
2a
equation of a vector:
v = 2ax + 2ay + 3az
describe with components:
2a,2a,3a
equivalent to:
2,2,3
v = 2,2,3éë ùû
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direction normal to a plane
x
y
z
2a
a
2a
(1, 2, 1) plane
v = 1,2,1éë ùû
Why is [1, 2, 1] normal to (1, 2, 1)?
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27
where it comes from
x
y
z
2a
a
2a
(1, 2, 1) plane
N = 1,2,1éë ùû
equation of a plane:
f x, y,z( ) = x
xint
+y
yint
+z
zint
= 1
normal to a plane:
N = Ñf x, y, z( ) = ¶ f
¶xx +
¶ f
¶yy +
¶ f
¶xz
N =
1
xint
x +1
yint
y +1
zint
z
(gradient)
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angle between planes
(1, 0, 0) plane
N
1= 1,0,0éë ùû
N
2= 1,1,1éë ùû
q
N
1·
N
2= N
1N
2cosq
(KOH etching)
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angle between planes
cosq =
N
1·
N
2
N1N
2
N
1= h
1,k
1,l
1éë ùû
N
2= h
2,k
2,l
2éë ùû
cosq =h
1h
2+ k
1k
2+ l
1l2
h1
2 + k1
2 + l1
2 h2
2 + k2
2 + l2
2
N
1= 1,0,0éë ùû
N
2= 1,1,1éë ùû
cosq =1+ 0+ 0
12 + 02 + 02 12
2 +12
2 +12
2
cosq =
1
3
q = 54.7
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summary
h k l( )
h k léë ùû
A specific plane.
A direction normal to the plane above.
h k l{ } A set of equivalent planes.
h k l A set of equivalent directions.
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what plane is this?
x
y
z
a
a
-a
a 2a 3a
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32
what plane is this?
y
x
z
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Silicon: atoms / cm2 on (100)
https://nanohub.org/tools/crystal_viewer
Lattice constant: 5.4307 Ang
Atoms on face = 4 times ¼ +1 = 2
Ns = 2/a2
Ns = 6.81x 1014 /cm2
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outline
1. Graphene
2. Silicon
3. Miller indices
4. Quantization of energy levels
5. Energy bands
6. Electrons and holes
7. Intrinsic carriers
8. Doping
1/11/2018
✔✔✔
35
silicon energy levels
1s2
2s2
2p6
3s2
3p2
4s0
4 valence electrons8 valence states
“core levels”
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energ
y
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36
quantization of energy levels
+
n = 1 n = 2
n = 3
E
H= -
13.6
n2eV n = 1,2,3,...
Hydrogen atom
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Bermel ECE 305 S18 37
outline
1. Graphene
2. Silicon
3. Miller indices
4. Quantization of energy levels
5. Energy bands
6. Electrons and holes
7. Intrinsic carriers
8. Doping
1/11/2018
✔✔✔✔
38
silicon energy levels energy bands
Si crystal
Bermel ECE 305 S18
3S2
3P2
energ
y
4Natoms statesconduction “band”
valence “band” 4Natoms states
“forbidden gap”
T = 0 K
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39
silicon energy levels energy bands
Si crystal
Bermel ECE 305 S18
3S2
3P2
energ
y
4Natoms statesconduction “band”
valence “band” 4Natoms states
“forbidden gap”
• • • • • • • • •
• • • • • • • • •
E =
3
2k
BT = 0.026 eV
T = 300 K
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40
energy band diagrams
Bermel ECE 305 S18
valence “band”: highestfilled band
“forbidden gap”
• • • • • • • • • • • • •
• • • • • • • • • • • • • EC
EV
• • • • • • • • •
• • • • • • • • •
E
G
p = nicm-3
n = p = nicm-3
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Conduction band: first partially empty band above filled valence band
energ
y
Si atoms1s22s22p63s22p2
C atoms1s22s22p2
2s – 2 states2p – 6 states
For N atoms:
2s line –2N-fold degenerate
2p line –6N-fold degenerate
energy bands versus atomic separation
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Empty states
Energy
42
insulators metals semiconductors
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do conduct electricity well
don’t conduct electricity well
in-between, but can be controlled
EC
E
C
EV E
V
E
G» 9 eV (SiO
2)
E
G»1.1eV (Si)
Filled states
Bermel ECE 305 S18 43
outline
1. Graphene
2. Silicon
3. Miller indices
4. Quantization of energy levels
5. Energy bands
6. Electrons and holes
7. Intrinsic carriers
8. Doping
1/11/2018
✔✔✔✔✔
summary
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1. Most solid materials are crystals, which fill space with periodically repeated elements
2. Showed how atomic energy level quantization leads to energy band formation in materials
3. Three types of materials: insulators, metals, and semiconductors
