Ionization energy – Bohr model
2
2
m* v
e
Fcentrifugal 
+ FCoulomb 
2
ra
ra
4 ra
2
2
e
m* v
 FCoulomb  Fcentrifugal

2
ra
4 ra
 m*  ra v
m* ra v 

e


m* ra
4
m* ra
angular momentum is quantized  m* rav  n
2 2
n 4
radius is quantized  ra 
2
m*
e
2
2
ra n m0
4
Bohr radius  a0 


2
a0
m*
m0 e
2
2
2 2
2
 n 2

m* ra
Ionization energy – Bohr model
total energy = kinetic energy + potential energy
2
2 2
1
e
n 4
2
 m* v 
m* ra v  n ra 
2
2
4 ra
m* e
1
 n 
 m* 

2
 m* ra 
2

e2
 n2 2 4 
4 
2 
 m* e 
1
n

 m* 
2
 n 2 2 4
 m* 
2
 m* e

2
e



2 2



n
4

  4 
2 

 m* e 
2
Ionization energy – Bohr model
2
1
n
e


 m* 


2
 n2 2 4 
 n2 2 4
 m* 
 4 

2
2
 m* e  
 m* 4e

4
4
1 m* e
m* e
1 m* e



2
2
2
2
 n 4 
2  n 4   n 4 
2
Ec
Ev



(3.37)
Fermi-Dirac distribution
1
f ( E )  EE 
F
 BT
e
1
1
1  f ( E )  1   EE 
F
 BT
e
1
 E  EF 

e
 BT
 E  EF 
e
11
 BT
 E  EF 
e
 BT
  EE 
F
 BT
1
e
1
Distribution of states & electrons
Fermi function
Ec
Ev
Ev
.
Ec
f Fermi ( E )
1
.
Ev
0
p

gv ( E )1  f Fermi ( E ) dE
n  E gc ( E ) f Fermi ( E )dE
c
Ev
Ec
EFermi
Figure 3-1
n( E )  gc ( E ) f Fermi ( E )
a
Figure 3-1
p( E )  gv ( E )1  f Fermi ( E )
a
Problem 3.5
The magnitude of the product g(E) & Fermi(E) in the
conduction band is a function of energy as shown in Figure
3.1. Assume the Boltzmann approximation is valid.
Determine the energy with respect to Ec at which the
maximum occurs.
n( E )  gc ( E ) f Fermi ( E )
   E  EF  
 E  Ec exp 

  BT

   E  Ec  
   Ec  EF  
 E  Ec exp 
 exp 

 BT
  BT 


n( E )  E  Ec
Problem 3.5
   E  Ec  
   Ec  EF  
exp
exp


 BT




 BT

   E  Ec   
d  E  Ec exp 

d  n( E )

  BT    0

dE
dE
x   E  Ec 

  x  
d  x exp 



  BT  
0
dx


Problem 3.5

  x  
d  x exp 



  BT  
0
dx
 1
 x 
x 


 exp 
 0
 2 x  BT 
  BT 
x   E  Ec 
 E  Ec 
 BT
2
Typical numbers
Normalized electron Normalized hole
effective mass
effective mass
Silicon
1.08
0.56
Gallium arsenide
0.067
0.48
Germanium
0.55
0.37
Normalized mass equals effective
mass/rest mass of the electron
n type extrinsic semiconductor
Ec
* 

mp
1
3
EFi   Ec  Ev    BT ln  * 
2
4
 mn 
donor energy level
Ev
Fermi energy between donor energy
band and conduction band
n( E )  gc ( E ) f Fermi ( E )
p( E )  gv ( E )1  f Fermi ( E )
p type extrinsic semiconductor
*

m
1
3
p
EFi   Ec  Ev    BT ln  *
2
4
 mn
acceptor energy level
Fermi energy between acceptor



Ec
Ev
energy band and conduction band
n( E )  gc ( E ) f Fermi ( E )
p( E )  gv ( E )1  f Fermi ( E )
densities may differ by 1010
Ionization energies in silicon
and germanium in electron volts
Impurity
Donors
Phosphorus
Arsenic
Si
Ge
0.045
0.05
0.012
0.0127
0.045
0.06
0.0104
0.0102
Acceptors
Boron
Aluminum
Conduction band
Valence band
Current density – charges in motion
J   vdrift A
-
J hole
2
m
 epvhole drift
e  1.602  10
19
C
J electron  envelectron drift
+
equation of motion for a hole in an electric field
F  m p * a  eE
mobility velectron drift   electron E & vhole drift  hole E
vn   n E
vp   pE
J electron  en( n E ) J hole  ep p E
J  e  p p  nn  E
collisions
+
equation of motion for a hole in an electric field
dv
F  mp *
 eE
dt
starting velocity after each collision is 0
eE
v
t
mp *
average time between collisions  cp
collisions
+
starting velocity after each collision is 0
eE
v
t
mp *
average time between collisions  cp
average velocity statistical analysis yields
eE
v
 cp
2m p *
eE
v
 cp
mp *
collisions
+
statistical analysis yields
hole mobility
eE
v
 cp
mp *
e
p 
 cp
mp *
e
electron mobility n 
 cn
mn *
Lattice structure will also vibrate
due to T > 0
“lattice scattering”
lattice mobility lattice
1
 
T 
3
2
Conductivity and resistivity
J  e  p p  nn  E
J E
  e  p p  nn  S m or mhos m
1
1
 
m
 e  p  p  n n 
Conductivity and resistivity
J E
I
J
A
V
V
E
L
I
You enter the laboratory and see an
experiment. How will you know which class is
it?
If it's green and wiggles, it's biology.
If it stinks, it's chemistry.
If it doesn't work, it's physics.
Teachers' remarks that changed the history of physics
Copernicus, when will you understand that you are not the center
of the world?
Galileo, if you will drop stones from the top of the tower one more
time, you will be dismissed forever.
Kepler, till when will you stare at the sky?
Newton, will you please stop idling away under the apple tree?
Ohm, must you resist Ampere's opinions on current events?
Nikola Tesla, I see that everyone is attracted to your magnetic
personality.
Einstein, a crocodile is greener or is it wider?
Schrödinger, stop abusing cats!
Heisenberg, when will you be sure of yourself?
Simple three-dimensional unit cell
c
b
a