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An overview of crystal structural properties, including miller indexes for crystalline directions and planes, spacing of equivalent planes, and bragg's law. It also covers energy gaps in solids, derived from the scattering of periodic potentials, and discusses the atomic origins of energy bands and their relation to metals, insulators, and semiconductors.
Typology: Schemes and Mind Maps
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that the final exam will be limited to the contents covered here. It is important to
review all HWs, sim labs, quizzes, previous exams, and other course materials.
Crystal Structural Properties
, 1 4 , 3 , 2
2 2 2
0
h k l
a
d
2 d sin n , n 1 , 2 , 3 ,
1/
1
1
1/
Crystal Planes and Directions
Crystal Planes and Directions
t
i
x t x e
( , ) ( )
( ) ( ) ( )
( )
V x x E x
dx
d x
m
2
2 2
2
t
x t
x t V x t i
dx
d
m
( , )
( , ) ( , )
2
2 2
2
( x , t ) ( x ) w ( t )
group particle
particle
phase
Electron in a Quantum Well
2
2
2
2
2
mE
k
k x
dx
d x
0
2
2
2
2
2
mV E
x
dx
d x
2
2
2
2
2
mV E
x
dx
d x
ikx ikx
x x
x x
( x ) finite 1
0 0
1 2
1 0 2 0
x x x x
x x
x x x x
( ) ( )
( a ) ik ( a ) ik ( a )
( a ) ik ( a ) ik ( a )
ika ika a
ika ika a
0
Electron in a Quantum Well
2
2
2
2
2 2
2
mE
k
k x
dx
d x
2
2
1
2
1 2
2
mE V
k
k x
dx
d x
ikx ikx
ik x ik x
1 1
ik x ikx
1 1
( x ) finite 1
0 0
1 2
1 0 2 0
x x x x
x x
x x x x
( ) ( )
( ) ( ) ( ) ( ) 1 1 2 2 ik a ik a ik a ik a
ika ika ika ika
1 1
0
2
2
1
2
1 2
2
m E V
k
k x
dx
d x
Yes. But not useful.
( )
2
( )
2
( )
1
( )
1
1 1 2 2
ik a ik a ik a ik a
ika ika ika ika
1 1
1 1
By adding symmetry requirement and the normalization condition, all parameters can be determined.
Potential: Periodic
Wavefunction: Bloch Theorem
Key conclusions: i) There are
bandgaps!!!! ii) 2N electronic
states in a band
Dispersion relation
Brillouin Zones (optional)
Electron effective mass
Explanation of insulators and
metals
Holes
0
DOS Distribution Functions Occupied DOS
3 1/
2 3
g E ( ) (2 m E )
1
1
exp[( )/ ]
( , )
E E k T
f E T
F B
( )
( ) ( )
exp[( ) / ] 1 F B
g E
g E f E
E E k T
1
1
1
exp[( )/ ]
( , )
E E k T
f E T
F B
*3 1/
2 3
c e g
*3 1/
2 3
v h v
g E m E E
ρ(E)
E
E c = E g
E v
= 0
g c (E)
g v (E)
Eg
1
1
exp[( )/ ]
( , )
E E k T
f E T
F B
E c
E v
HOLES
ELECTRONS
0.
E F
E
f ( E )
0 K
600 K
6000 K
(^01)
( )
( ) ( ) ( )
exp[( ) / ] 1
occ (^) e
e e
F B
g E
g E g E f E
E E k T
exp[( ) / ] 1
occ
v v
v
F B
g E g E f E
g E
E E k T
( ) ln
e
h
B
g
F
m
m
k T
)* 13. 6 ( )
( 2
0
eV
m
m
E E
r
c D
0 0
0 D A
p n N N
D D D
A A A
Number of ionized acceptors
2
0 0
exp
g
i c v
B
E
n p n N N
k T
0 0
g
i c v
B
**For HEAVY doping and FULL ionization of dopants, the position of the Fermi energy in** the gap varies as
A D i
i
A
F i B
E E k T
ln ,
D A i
i
D
F i B
E E k T
ln ,
D D A A
1/ 2 2
2
0
0
2 2
i A D A D
i
n N N N N
p p n
n
(^)
1/ 2
2
0
2 2
D A D A
i
N N N N
n n N
^
(^)
3/
( )/
(^0 )
1 2
,
4
F c B E E k T (^) e B
c c
m k T
n n N e N
3/ 2
( )/
(^0 )
1 2
,
4
v F B E E k T (^) h B
v v
m k T
p p N e N
e edrift ediffusion e e e e
h hdrift hdiffusion h h h h
e N eD N
e N eD N
J J J E
J J J E
h diff h h ediff e^ e
J eD N J eD N
h drift hdiffusion edrift e diffusion
n p
n p
Potential
energy
-x x p x n
p-TYPE n-TYPE
DEPLETION
REGION
V(x)
2
2
B A D D A
bi
i D A
e n N N
N d
-N a
-x p
x n
- - -
+ + +
E v
E c FERMI LEVEL E F
p-TYPE
REGION
n-TYPE
REGION
eV bi
a p d n
N x N x
ChargeNeutralityRequires :
A
x p
D
x n
1/
bi
A D
e N N