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Lecture 5
PN Junctions in Thermal Equilibrium
In this lecture you will learn:
• Junctions of P and N doped semiconductors
• Electrostatics of PN junctions in thermal equilibrium
• Built-in junction potential, junction electric field, depletion regions, and
all that…
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
Potential of Doped Semiconductors
What are the values of potentials in N-doped and P-doped semiconductors ??
N-doped Semiconductors (doping density is Nd ):
The potential in n-doped semiconductors is denoted by: n
no  x   Nd
 Nd  ni e
 n 
q n  x 
KT
N 
KT
log d 
q
 ni 
Example:
Suppose,
Nd  1017 cm- 3 and ni  1010 cm-3
 n 
N 
KT
log d    0.41 Volts
q
 ni 
P-doped Semiconductors (doping density is Na ):
The potential in n-doped semiconductors is denoted by: p
po  x   Na
 Na  ni e
 p  

q p  x 
KT
N 
KT
log a 
q
 ni 
Example:
Suppose,
Na  1017 cm- 3 and ni  1010 cm-3
 p  
N 
KT
log a    0.41 Volts
q
 ni 
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
1
Can One Measure the Potential Difference
between Doped Semiconductors Directly?
P-doped
Na
p
N-doped
n
Nd
Answer: No!
For the explanation, you need to wait till the end of this handout…….
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
A Junction of P and N Doped Semiconductors
What do we know (before the junction is formed):
P-doped
N-doped
Na
Nd
Doping:
Doping:
Na
Nd
Carrier densities:
Carrier densities:
nno  Nd
p po  N a
n po 
Potential:
ni2
Na
pno 
N 
KT
  p  
log a 
q
 ni 
Potential:
  n 
ni2
Nd
N 
KT
log d 
q
 ni 
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
2
Step 1: Diffusion of Electron and Holes
Metal
contact
P-doped
N-doped
Na
Nd
0
Metal wire
x
Carrier diffusion
Holes are more in number on the P-side, so they will diffuse into the N-side from the
P-side
--- On reaching the N-side, the hole will find an electron and recombine with it
Electrons are more in number on the N-side, so they will diffuse into the P-side from
the N-side
--- On reaching the P-side, the electron will find a hole and recombine with it
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
Step 2: Creation of “Depletion” Regions
P-doped
Na
- - - - - - - - - -
+ + + +
+ + + +
+ + + +
N-doped
Nd
0
x
Region depleted of holes
Region depleted of electrons
Charge density:  qN a
Charge density:  qN d
Both the N-doped and P-doped materials were charge neutral before the junction was
formed
As the holes diffuse from the P-side into the N-side, they leave behind negatively
charged acceptor atoms in a region near the interface on the P-side which becomes
“depleted” of holes
As the electrons diffuse from the N-side into the P-side, they leave behind positively
charged donor atoms in a region near the interface on the N-side which becomes
“depleted” of electrons
As the carrier keep diffusing, the widths of the depletion regions keep increasing ……..
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
3
Generation of Electric Field
Electric field
- - - - - - - - - -
P-doped
Na
+ + + +
+ + + +
+ + + +
N-doped
Nd
0
x
Region depleted of holes
Region depleted of electrons
Charge density:  qN a
Charge density:  qNd
The charge densities in the depletion regions generate an electric field
As the depletion regions grow in thickness, the magnitude of the electric field also
increases
Question: How do we figure out how big this electric field is?........
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
Generation of Electric Field
Electric field
- - - - - - - - - -
P-doped
Na
 x po
+ + + +
+ + + +
+ + + +
0
N-doped
Nd
x no
x
Region depleted of holes
Region depleted of electrons
Charge density:  qN a
Charge density:  qNd
Assume that the thickness of the depletion region on the P-side is: x po
Assume that the thickness of the depletion region on the N-side is: x no
Gauss’s Law on the P-side:
 E  x po   0
qN a
dE

s
dx
 E x   
Gauss’s Law on the N-side:
qNa
s
x  x po 
qNd
dE

dx
s
 E x   
 E  x no   0
qNd
s
 x no  x 
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
4
Generation of Electric Field
Electric field
- - - - - - - - - -
P-doped
Na
 x po
+ + + +
+ + + +
+ + + +
Nd
x no
0
x
Region depleted of holes
Region depleted of electrons
Charge density:  qN a
Charge density:  qN d
Field on the P-side:
E x   
N-doped
qN a
s
Field on the N-side:
x  x po 
E x   
E x 
0
 x po
qNd
s
 x no  x 
x
x no
Field varies linearly with distance
The field must be continuous at the junction: qN a x po  qNd x no
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
Step 3: Establishment of Equilibrium
Electric field
- - - - - - - - - -
P-doped
Na
 x po
+ + + +
+ + + +
+ + + +
0
N-doped
Nd
x no
Region depleted of holes
Region depleted of electrons
Charge density:  qN a
Charge density:  qNd
x
As the carriers diffuse, and the depletion regions grow in thickness, and the electric
field increases, the drift currents due to the electric field also increase …………
A stage is reached when the carrier diffusion currents are exactly balanced by the
(oppositely directed) drift currents:
---- The electron diffusion current is balanced by the equal and opposite electron
drift current
---- The hole diffusion current is balanced by the equal and opposite hole drift
current
J p  x   J pdrift  x   J pdiff  x   0
J n  x   J ndrift  x   J ndiff  x   0
So the net currents of both the electrons as well as the holes go to zero …. and
equilibrium is established!!
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
5
A PN Junction in Equilibrium
Electric field
- - - - - - - - - -
P-doped
Na
 x po
+ + + +
+ + + +
+ + + +
N-doped
Nd
x no
0
x
Region depleted of holes
Region depleted of electrons
Charge density:  qN a
Charge density:  qNd
In equilibrium, there is no NET current (of either electrons or the holes)
But, in equilibrium, there is an electric field…….right at the junction!
E x 
0
 x po
x
x no
A non-zero electric field implies a spatially varying electrostatic potential:
E x   
d  x 
dx
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
A PN Junction in Equilibrium
Electric field
- - - - - - - - - -
P-doped
Na
 x po
+ + + +
+ + + +
+ + + +
N-doped
Nd
x no
0
x
A non-zero electric field implies an electrostatic potential:
E x 
 x po
0
E x   
Potential on the P-side:
N 
KT
  p  
log a 
q
 ni 
x
x no
d  x 
dx
Potential on the N-side:
  n 
N 
KT
log d 
q
 ni 
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
6
A PN Junction in Equilibrium
Electric field
P-doped
Na
- - - - - - - - - -
  p
+ + + +
+ + + +
+ + + +
 x po
N-doped
  n
Nd
x no
0
x
A non-zero electric field implies an electrostatic potential:
E x 
E x   
qN a
s
x  x po 
0
 x po
x no
d  x 
 E  x 
dx
qN a
x  x po 2
  x   p 
2 s
E x   
qNd
s
x
 x no  x 
d  x 
 E  x 
dx
qNd
 x no  x 2
   x   n 
2 s
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
A PN Junction in Equilibrium
Electric field
P-doped
Na
- - - - - - - - - -
  p
+ + + +
+ + + +
+ + + +
 x po
 x   p 
qN a
x  x po 2
2 s
0
N-doped
  n
Nd
x no
x
n 
 x 
N 
KT
log d 
q
 ni 
n
B
 x po
p
p  
0
x
x no
B  n   p
N 
KT
log a 
q
 ni 
  x   n 
qNd
 x no  x 2
2 s
Potential varies quadratically with distance
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
7
The Built-In Potential
Electric field
P-doped
- - - - - - - - - -
  p
Na
 x po
+ + + +
+ + + +
+ + + +
N-doped
  n
Nd
x no
0
x
 x 
 x po
p
n
0
B
x no
x
B  n   p
Built-In Junction Potential:
Example:
Nd  10171/cm3
B  n   p

N a  10171/cm3
B  n   p
N N 
KT
log d 2 a 
q
 ni 

N N 
KT
log d 2 a   0.83 Volts
q
 ni 
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
A PN Junction in Equilibrium: Depletion Region Widths
Electric field
P-doped
Na
- - - - - - - - - -
  p
 x po
+ + + +
+ + + +
+ + + +
0
N-doped
  n
Nd
x no
x
The electric field and the potential must be continuous at the junction
Field and Potential on the P-side:
E x   
 x   p 
qN a
s
Field and Potential on the N-side:
x  x po 
E x   
qN a
x  x po 2
2 s
qNd
s
  x   n 
 x no  x 
qNd
 x no  x 2
2 s
qN a x po  qNd x no
qN a
2
d x
x po 2  n  qN
no 
2 s
2 s
These two equations can be solved to find xpo and xno
p 
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
8
A PN Junction in Equilibrium: Depletion Region Widths
Electric field
P-doped
Na
- - - - - - - - - -
  p
+ + + +
+ + + +
+ + + +
 x po
N-doped
  n
Nd
x no
0
x
The result is:
2  s B
qN a
x po 
 Nd

 N a  Nd



x no 
2  s B
qNd
 Na

 N a  Nd



Observations:
i)
If Nd >> Na:
x po 
ii)
If Na >> Nd:
x po 
2  s B
2  s B
 x no 
qN a
qNd
2  s B
qN a
 Nd

 Na
 Na

 Nd

2  s B
  x no 
qNd




In an asymmetrically
doped junction, the
depletion region on
the lightly doped
side is larger
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
A PN Junction in Equilibrium: A Summary
Electric field
P-doped
Na
- - - - - - - - - -
  p
+ + + +
+ + + +
+ + + +
 x po
0
N-doped
  n
Nd
x no
x
E x 
0
 x po
Emax 
2 q B  N d N a

 s  N a  Nd



x
x no
 x 
n
B
p
 x po
0
x
x no
B  n   p
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
9
A PN Junction Process
P-dopant
(Ion implantation)
P-doped
A Si wafer (N-doped)
A Si wafer (N-doped)
Metal
Metal
Metal
P-doped
P-doped
A Si wafer (N-doped)
A Si wafer (N-doped)
Depletion region
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
Can One Measure the Potential Difference
between Doped Semiconductors Directly?
P-doped
M
M
p
n
N-doped
Metals also have a
potential: M
There is contact potential at the metal-semiconductor interface
Contact potential is just the built-in potential at the metal-semiconductor interface
After accounting for the contact-potential, the potential difference measured by the
voltmeter is zero!
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
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