Lundstrom ECE-305 S15
ECE-305: Spring 2015
Material Properties: I
Professor Mark Lundstrom Electrical and Computer Engineering
Purdue University, West Lafayette, IN USA [email protected]
1/14/15
Pierret, Semiconductor Device Fundamentals (SDF) pp. 3-19
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https://nanohub.org/groups/ece305lundstrom
https://nanohub.org/groups/ece305lundstrom
semiconductors
“One shouldn’t work on semiconductors, that is a filthy mess; who knows whether any semiconductors exist.” Wolfgang Pauli, 1931
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outline
1. Silicon
2. Crystals
3. Miller indices
semiconductors
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http://en.wikipedia.org/wiki/Periodic_table
column IV
6
silicon energy levels
1S2
2S2
2P6
3S2
3P2
4S0
Si atom (At. no. 14)
4 valence electrons 8 valence states
“core levels”
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ener
gy
Si atoms in a solid
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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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a 2D crystal
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a
b
Graphene: 2011 Nobel Prize in Physics
9
Graphene is a one-atom-thick planar carbon sheet with a honeycomb lattice.
source: CNTBands 2.0 on nanoHUB.org
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unit cells
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lattice plus basis
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3D crystal structure
12 http://en.wikipedia.org/wiki/Bravais_lattice
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semiconductors
13
http://en.wikipedia.org/wiki/Periodic_table
column IV
semiconductors
14
http://en.wikipedia.org/wiki/Periodic_table
Col. III
Col. V
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Si crystal structure (diamond lattice)
Natoms ≈ 5×1022 cm-3
5.43 A
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4 nearest neighbors
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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: density
https://nanohub.org/tools/crystal_viewer
Lattice constant: a = 5.4307 Ang Density = total mass/vol. of unit cell. Atomic mass of Si: 28.0855 amu 1 amu = 1.6605 x 10−27 kg
ρ = 8× 28.0855×1.6605×10−27
5.4307 ×10−10( )3 kg/m3
ρ = 2.3296 g/cm3
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Silicon: NN spacing
https://nanohub.org/tools/crystal_viewer
Lattice constant: a = 5.4307 Ang Body diagonal = sqrt(3) a. NN spacing = sqrt(3)a/4
“cartoon” Si crystal
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outline
1. Silicon
2. Crystals
3. Miller indices
✔
✔
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Miller index prescription for describing planes
x
y
z
2a
a
2a
x, y, and z-axis intercepts: 2a, 1a, 2a 2, 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:
xxint
+ yyint
+ zzint
= 1
describe with the numbers:
1xint
, 1yint
, 1zint
equivalent to:
1xint a
, 1yint a
, 1zint a
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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 = 2,2,3⎡⎣ ⎤⎦
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direction normal to a plane
x
y
z
2a
a
2a
(1, 2, 1) plane
v = 1,2,1⎡⎣ ⎤⎦
Why is [1, 2, 1] normal to (1, 2, 1)?
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where it comes from
x
y
z
2a
a
2a
(1, 2, 1) plane
N = 1,2,1⎡⎣ ⎤⎦
equation of a plane:
f x, y, z( ) = x
xint
+ yyint
+ zzint
= 1
normal to a plane:
N = ∇f x, y, z( ) = ∂ f
∂xx + ∂ f
∂yy + ∂ f
∂xz
N = 1
xint
x + 1yint
y + 1zint
z
(gradient)
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angle between planes
(1, 0, 0) plane
(1, 1, 1) plane
N1 = 1,0,0⎡⎣ ⎤⎦
N2 = 1,1,1⎡⎣ ⎤⎦
θ
N1 •
N2 = N1N2 cosθ
(KOH etching)
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angle between planes
cosθ =
N1 •
N2
N1N2
N1 = h1,k1,l1⎡⎣ ⎤⎦
N2 = h2 ,k2 ,l2⎡⎣ ⎤⎦
cosθ =
h1h2 + k1k2 + l1l2
h12 + k1
2 + l12 h2
2 + k22 + l2
2
N1 = 1,0,0⎡⎣ ⎤⎦
N2 = 1,1,1⎡⎣ ⎤⎦
cosθ = 1+ 0+ 0
12 + 02 + 02 122 +12
2 +122
cosθ = 1
3
θ = 54.7
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summary
h k l( )
!N = hax + kay + laz h k l⎡⎣ ⎤⎦
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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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
https://nanohub.org/tools/crystal_viewer Online tool:
1. Silicon
2. Crystals
3. Miller indices
