Lecture #15
OUTLINE
• pn junction I-V characteristics
Reading: Chapter 6.1
NOTE:
• Typically, pn junctions in IC devices are formed by
counter-doping. The equations derived in class (and in
the textbook) can be readily applied to such diodes if
NA  net acceptor doping on p-side (NA-ND)p-side
ND  net donor doping on n-side (ND-NA)n-side
Spring 2007
EE130 Lecture 15, Slide 1
Linearly Graded Junction
Spring 2007
EE130 Lecture 15, Slide 2
Biased PN Junctions
Note that VA should be significantly smaller than Vbi
(Otherwise, we cannot assume low-level injection)
Spring 2007
EE130 Lecture 15, Slide 3
Effect of Bias on Electrostatics
Spring 2007
EE130 Lecture 15, Slide 4
pn Junction Electrostatics, VA  0
• Built-in potential Vbi (non-degenerate doping):
kT  N A  kT  N D  kT  N A N D 

 
 
Vbi 
ln 
ln 
ln 
2
q  ni  q  ni  q  ni 
• Depletion width W :
 1
2 s
1 

W  x p  xn 
(Vbi  VA )

q
 N A ND 
ND
xp 
W
N A  ND
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NA
xn 
W
N A  ND
EE130 Lecture 15, Slide 5
• Electric field distribution
(x)
• Potential distribution V(x)
ND
Note that V (0) 
(Vbi  VA )
N A  ND
Spring 2007
EE130 Lecture 15, Slide 6
Peak Electric Field
1
  dx  2  (0) W  Vbi  VA
• For a one-sided junction: W 
2 s
(Vbi  VA )
qN
2Vbi  VA 
2qN Vbi  VA 

therefore  (0) 
W
s
Spring 2007
EE130 Lecture 15, Slide 7
Current Flow - Qualitative
Spring 2007
EE130 Lecture 15, Slide 8
Current Flow in a pn Junction Diode
• When a forward bias (VA>0) is applied, the
potential barrier to diffusion across the
junction is reduced
– Minority carriers are “injected” into the quasineutral regions => Dnp > 0, Dpn > 0
• Minority carriers diffuse in the quasi-neutral
regions, recombining with majority carriers
Spring 2007
EE130 Lecture 15, Slide 9
• Current density J = Jn(x) + Jp(x)
dn
d ( Dn )
J n ( x )  qn n  qDn
 qn n  qDn
dx
dx
dp
d ( Dp )
J p ( x )  q p p  qD p
 q p p  qD p
dx
dx
• J is constant throughout the diode, but Jn(x)
and Jp(x) vary with position
Spring 2007
EE130 Lecture 15, Slide 10
Ideal Diode Analysis: Assumptions
• Non-degenerately doped step junction
• Steady-state conditions
• Low-level injection conditions prevail in the
quasi-neutral regions
• Recombination-generation is negligible in the
depletion region
dJ n

 0,
dx
dJ p
dx
0
i.e. Jn & Jp are constant inside the depletion region
Spring 2007
EE130 Lecture 15, Slide 11
Ideal Diode Analysis: Approach
• Solve the minority-carrier diffusion equations in
quasi-neutral regions to obtain Dnp(x,VA),Dpn(x,VA)
– apply boundary conditions
• p-side: Dnp(-xp), Dnp(-)
• n-side: Dpn(xn), Dpn()
• Determine minority-carrier current densities in quasineutral regions
d ( Dn p )
d ( Dpn )
J p ( x,VA )   qD p
J n ( x,VA )  qDn
dx
dx
• Evaluate Jn at x=-xp and Jp at x=xn
J(VA) = Jn(VA)|x=-xp + Jp(VA )|x=xn
Spring 2007
EE130 Lecture 15, Slide 12
Carrier Concentrations at –xp, xn
Consider the equilibrium (VA = 0) carrier concentrations:
p-side
n-side
p p 0 ( x p )  N A
nn 0 ( xn )  N D
2
i
n
n p 0 ( x p ) 
NA
2
i
n
p n 0 ( xn ) 
ND
If low-level injection conditions prevail in the quasi-neutral
regions when VA  0, then
p p ( x p )  N A
Spring 2007
nn ( xn )  N D
EE130 Lecture 15, Slide 13
“Law of the Junction”
The voltage VA applied to a pn junction falls mostly across
the depletion region (assuming that low-level injection
conditions prevail in the quasi-neutral regions).
We can draw 2 quasi-Fermi levels in the depletion region:
p  ni e( Ei  FP ) / kT
n  ni e( FN  Ei ) / kT
pn  ni2e( Ei  FP ) / kT e( FN  Ei ) / kT
 ni2e( FN  FP ) / kT
pn  ni2eqVA / kT
Spring 2007
EE130 Lecture 15, Slide 14
Excess Carrier Concentrations at –xp, xn
p-side
n-side
p p ( x p )  N A
nn ( xn )  N D
ni2 e qVA / kT
n p ( x p ) 
NA
ni2 e qVA / kT
p n ( xn ) 
ND
 n p 0 e qVA / kT
2
i

 pn 0 e qVA / kT

n
Dn p ( x p ) 
e qVA / kT  1
NA
Spring 2007
2
i


n
Dpn ( xn ) 
e qVA / kT  1
ND
EE130 Lecture 15, Slide 15
Example: Carrier Injection
A pn junction has NA=1018 cm-3 and ND=1016 cm-3. The applied
voltage is 0.6 V.
Question: What are the minority carrier concentrations at the
depletion-region edges?
qV
Answer: n p ( x p )  n poe A
pn ( xn )  pnoe qVA
kT
kT
 100  e0.6 0.026  1012 cm-3
 104  e0.6 0.026  1014 cm -3
Question: What are the excess minority carrier concentrations?
Answer:
Dn p ( x p )  n p ( x p )  n po  1012 100  1012 cm-3
Dpn ( xn )  pn ( xn )  pno  1014  104  1014 cm -3
Spring 2007
EE130 Lecture 15, Slide 16
Excess Carrier Distribution
d 2 Dpn
Dpn
Dpn


dx 2
Dp p Lp 2
• From the minority carrier diffusion equation:
• We have the following boundary conditions:
Dpn ( xn )  pno (e qVA / kT  1)
Dpn ()  0
• For simplicity, we will develop a new coordinate system:
NEW:
x’’
0
0
x’
• Then, the solution is of the form:
Dpn ( x' )  A1e
Spring 2007
x '/ L p
 A2e
EE130 Lecture 15, Slide 17
 x '/ L p
Dpn ( x' )  A1e
x ' / Lp
 A2e
 x ' / Lp
From the x =  boundary condition, A1 = 0.
qVA / kT
A

p
(
e
 1)
From the x = xn boundary condition, 2
no
Therefore, Dpn ( x' )  pno (e
qVA / kT
 1)e
 x ' / Lp
, x'  0
Similarly, we can derive
Dn p ( x' ' )  n po (eqVA / kT 1)e x ''/ Ln , x' '  0
Spring 2007
EE130 Lecture 15, Slide 18
pn Diode I-V Characteristic
dDn p ( x' ' )
p-side: J n  qDn
dx' '
Dn
q
n p 0 (e qVA
Ln
Dp
dDpn ( x' )
qVA
n-side: J p   qD p
q
pn 0 (e
dx'
Lp
J  J n x  x  J p
p
x  xn
 J n x0  J p
 Dn
D p  qVA
J  qn 

( e
 Ln N A Lp N D 
2
i
Spring 2007
EE130 Lecture 15, Slide 19
kT
x  0
 1)
kT
 1)e  x '' Ln
kT
 x' Lp
 1)e
I  I 0 (e qVA
kT
 1)
 Dp

D
n

I 0  Aqni 

L N

L
N
p
D
n
A


2
Spring 2007
EE130 Lecture 15, Slide 20
Diode Saturation Current I0
 Dp

D
n

I 0  Aqni 

L N

L
N
p
D
n
A


2
• I0 can vary by orders of magnitude, depending on the
semiconductor material
• In an asymmetrically doped pn junction, the term
associated with the more heavily doped side is negligible:
 Dp 

– If the p side is much more heavily doped, I 0  Aqni 
L N 
 p D
2
 Dn 

– If the n side is much more heavily doped, I 0  Aqni 
 Ln N A 
2
Spring 2007
EE130 Lecture 15, Slide 21
Summary
•
The total voltage dropped across a pn junction is Vbi-VA:
2 s Vbi  VA   1
1 


– Depletion-layer width W 

q
 N A ND 
2Vbi  VA 
– Peak electric field (0) 
W

•
Under forward bias (VA > 0), the potential barrier to
carrier diffusion is reduced
 minority carriers are “injected” and diffuse in the
quasi-neutral regions
 Dn
D p  qVA

Diode current I  qAn 
 (e
 Ln N A L p N D 
2
i
Spring 2007
EE130 Lecture 15, Slide 22
kT
 1)