Lecture 12
P-N Junction Diodes: Part 2
How do they work? (A little bit of math)
Reading:
Pierret 6.1
Georgia Tech
ECE 3040 - Dr. Alan Doolittle
Movement of electrons and holes when forming the junction
Circles are charges free to move
(electrons and holes)
Squares are charges NOT free to
move (ionized donor or acceptor
atoms)
High hole
Concentration
Electron diffusion
Fig Pierret 5.5High electron
Concentration
Hole diffusion
Local region of
negative
charge due to
imbalance in
hole-acceptor
concentrations
Local region of
positive charge
due to
imbalance in
electron-donor
concentrations
Space Charge or Depletion Region
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ECE 3040 - Dr. Alan Doolittle
Movement of electrons and holes when forming the junction
E= - dV/dx
-Edx=dV
xn
V ( xn )
−xp
V (− x p )
− ∫ Edx = ∫
= V ( xn ) − V (− x p ) = Vbi
dV
but...
dn
J N = qµ n nE + qDN
=0
dx
dn
dn
D
kT dx
E = − N dx = −
µn n
q n
No net current flow
in equilibrium
thus...
kT
Vbi = − ∫ Edx =
−xp
q
xn
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∫
n ( xn )
n(− x p )
dn
dx = kT ln  n( xn ) 


n
q  n(− x p ) 
ECE 3040 - Dr. Alan Doolittle
Movement of electrons and holes when forming the junction


kT  n( x n )  kT  N D 
Vbi =
ln  2
ln 
=

q  n(− x p )  q  ni

 NA 
kT  N A N D 
Vbi =
ln 

q  ni2 
For NA=ND=1015/cm-3 in silicon at room temperature,
Vbi~0.6 V*
For a non-degenerate semiconductor, |-qVbi|<|Eg|
*Note to those familiar with a diode turn on voltage: This is not the diode turn on voltage!
This is the voltage required to reach a flat band diagram and sets an upper limit (typically an
overestimate) for the voltage that can be applied to a diode before it burns itself up.
Georgia Tech
ECE 3040 - Dr. Alan Doolittle
Movement of electrons and holes when forming the junction
Depletion Region Approximation
++
-------------------
+ + + + ++++++++++++++++
-
Depletion Region Approximation states that approximately no free carriers exist in
the space charge region and no net charge exists outside of the depletion region
(known as the quasi-neutral region). Thus,
ρ
dE
q
=
=
( p − n + N D − N A ) = 0 within the quasi − neutral region
dx K S ε o K S ε o
becomes...
dE
q
=
( N D − N A ) within the space ch arg e region
dx K S ε o
Georgia Tech
ECE 3040 - Dr. Alan Doolittle
Movement of electrons and holes when forming the junction
Depletion Region Approximation: Step Junction Solution
 − qN A for − x p ≤ x ≤ 0

ρ =  qN D for 0 ≤ x ≤ x n
0 for x ≤ − x and x ≥ x
p
n

Fig. 5.9 a and b
thus,
 − qN A
for − x p ≤ x ≤ 0
 K ε
 S o
dE  qN D
=
for 0 ≤ x ≤ x n
dx  K S ε o
0 for x ≤ − x p and x ≥ x n

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Where we have used:
ρ
dE
=
dx K S ε o
ECE 3040 - Dr. Alan Doolittle
Movement of electrons and holes when forming the junction
Depletion Region Approximation: Step Junction Solution
∫
E ( x)
0
dE ' = ∫
x
−xp
− qN A
dx'
KSεo
− qN A
(x + x p )
E ( x) =
KSεo
for − x p ≤ x ≤ 0
for − x p ≤ x ≤ 0
and
qN D
dx'
KSεo
for 0 ≤ x ≤ x n
− qN D
(xn − x )
E ( x) =
KSεo
for 0 ≤ x ≤ x n
∫
0
E ( x)
dE ' = ∫
xn
x
Since E(x=0-)=E(x=0+)
NAxp=NDxn
Georgia Tech
ECE 3040 - Dr. Alan Doolittle
Movement of electrons and holes when forming the junction
E=−
Depletion Region Approximation: Step Junction Solution
dV
dx
 qN A
(
x p + x ) for − x p ≤ x ≤ 0

dV  K S ε o
=
dx  qN D ( x − x ) for 0 ≤ x ≤ x
n
n
 K S ε o
or ,
V ( x)
∫
0
VBi
∫
V ( x)
dV ' = ∫
x
−xp
dV ' = ∫
xn
x
qN A
(x p + x')dx'
KSεo
for − x p ≤ x ≤ 0
qN D
(xn − x')dx'
KSεo
for 0 ≤ x ≤ x n
 qN A
2
(
)
+
x
x
for − x p ≤ x ≤ 0
p
 2 K ε
S o
V ( x) = 
qN D
Vbi −
(xn − x )2 for 0 ≤ x ≤ xn

2K S ε o
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V=VBi
V=0
ECE 3040 - Dr. Alan Doolittle
Movement of electrons and holes when forming the junction
Depletion Region Approximation: Step Junction Solution
At x=0,
qN A
(x p )2 = Vbi − qN D (xn )2
2K S ε o
2K S ε o
U sin g , x p =
xn =
2K S ε o
NA
Vbi
q N D (N A + N D )
W = x p + xn =
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(xn N D )
and
xp =
NA
2K S ε o
ND
Vbi
q N A (N A + N D )
2K S ε o (N A + N D )
Vbi
q
NAND
ECE 3040 - Dr. Alan Doolittle
Movement of electrons and holes when forming the junction
Depletion Region Approximation: Step Junction Solution
Vbi
VA=0 : No Bias
Vbi
VA<0 : Reverse Bias
|VA|
VA>0 : Forward Bias
|VA|
Vbi
Vbi
Georgia Tech
ECE 3040 - Dr. Alan Doolittle
Movement of electrons and holes when forming the junction
Depletion Region Approximation: Step Junction Solution
Thus, only the boundary conditions change resulting in
direct replacement of Vbi with (Vbi-VA)
xn =
2K S ε o
NA
(Vbi − V A ) and
q N D (N A + N D )
W = x p + xn =
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xp =
2K S ε o
ND
(Vbi − V A )
q N A (N A + N D )
2K S ε o (N A + N D )
(Vbi − V A )
q
NAND
ECE 3040 - Dr. Alan Doolittle
Movement of electrons and holes when forming the junction
Step Junction Solution: What does it mean?
Consider a p+ -n junction (heavily doped p-side, normal or lightly doped n side).
Georgia Tech
ECE 3040 - Dr. Alan Doolittle
Movement of electrons and holes when forming the junction
Step Junction Solution: What does it mean?
Fermi-level only
applies to equilibrium
(no current flowing)
Majority carrier
Quasi-fermi
levels
Efp-Efn=-qVA
Georgia Tech
ECE 3040 - Dr. Alan Doolittle