EE130/230A
Discussion 6
Peng Zheng
Carrier Action under Forward Bias
• When a forward bias (VA>0) is applied, the potential
barrier to diffusion across the junction is reduced
– Minority carriers are “injected”
into the quasi-neutral regions
=> Dnp > 0, Dpn > 0
• Minority carriers diffuse in the quasi-neutral regions,
recombining with majority carriers
EE130/230A Fall 2013
Lecture 10, Slide 2
Components of Current Flow
• 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:
Example:
p+n junction under forward bias:
J
JN
JP
-xp
EE130/230A Fall 2013
Lecture 10, Slide 3
xn
x
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
 pn 0 e qVA / kT
 n p 0 e qVA / kT
2
i


n
Dn p ( x p ) 
e qVA / kT  1
NA
EE130/230A Fall 2013


ni2 qVA / kT
Dpn ( xn ) 
e
1
ND
Lecture 10, Slide 4
Carrier Concentration Profiles
under Forward Bias
EE130/230A Fall 2013
Lecture 10, Slide 5
R. F. Pierret, Semiconductor Device Fundamentals, Fig. 6.8a
Excess Carrier Distribution (n side)
• From the minority carrier diffusion equation:
d 2 Dpn
Dpn
Dpn


dx 2
Dp p Lp 2
• We have the following boundary conditions:
Dpn ()  0
Dpn ( xn )  pno (e qVA / kT  1)
• For simplicity, use a new coordinate system:
NEW:
x’’
0
0
x’
• Then, the solution is of the form: Dp ( x' )  A e x '/ Lp  A e x '/ Lp
n
1
2
EE130/230A Fall 2013
Lecture 10, Slide 6
Dpn ( x' )  A1e
x ' / Lp
 A2e
 x ' / Lp
From the x =  boundary condition:
From the x = xn boundary condition:
 x ' / Lp
, x'  0
 x ''/ Ln
, x' '  0
Therefore Dpn ( x' )  pno (eqVA / kT  1)e
Similarly, we can derive
Dn p ( x' ' )  n po (e
EE130/230A Fall 2013
qVA / kT
1)e
Lecture 10, Slide 7
Total Current Density
p side:
dDn p ( x' ' )
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
EE130/230A Fall 2013
Lecture 10, Slide 8
kT
x  0
 1)
kT
 1)e  x '' Ln
kT
 x' Lp
 1)e
Summary: Long-Base Diode
• Under forward bias (VA > 0), the potential barrier to carrier
diffusion is reduced  minority carriers are “injected” into the
quasi-neutral regions.
– The minority-carrier concentrations at the edges of the depletion region
change with the applied bias VA, by the factor e qVA / kT
– The excess carrier concentrations in the quasi-neutral regions decay to
zero away from the depletion region, due to recombination.
 Dn
D p  qVA

pn junction diode current I  qAn 
 (e
 Ln N A L p N D 
2
i
kT
• I0 can be viewed as the drift current due to minority carriers
generated within a diffusion length of the depletion region
EE130/230A Fall 2013
Lecture 10, Slide 9
 1)
General Narrow-Base Diode I-V
• Define WP‘ and WN’ to be the widths of the quasi-neutral regions.
• If both sides of a pn junction are narrow (i.e. much shorter than
the minority carrier diffusion lengths in the respective regions):
 DP
DN  qVA / kT
qVA / kT
I  qAni 

e
1  I0 e
1

WN N D WP N A 
2



e.g. if hole injection
J
into the n side is greater
than electron injection
JN
into the p side:
JP
-xp
EE130/230A Fall 2013

Lecture 11, Slide 10
xn
x
Summary: Narrow-Base Diode
• If the length of the quasi-neutral region is much shorter than the
minority-carrier diffusion length, then there will be negligible
recombination within the quasi-neutral region and hence all of the
injected minority carriers will “survive” to reach the metal contact.
– The excess carrier concentration is a linear function of distance.
For example, within a narrow n-type quasi-neutral region:
pno (e
qVA / kT
Dpn(x)
 1)
0
xn
location of metal contact
(Dpn=0)
x
WN’
 The minority-carrier diffusion current is constant within the narrow quasi-neutral
region.
Shorter quasi-neutral region  steeper concentration gradient  higher diffusion current
EE130/230A Fall 2013
Lecture 11, Slide 11
Sample Problem
Consider a Si pn step junction diode maintained at room temperature, with p-side and n-side
dopant concentrations NA = 1016 cm-3 and ND = 21016 cm-3, respectively. (You may assume
that each side is uncompensated.) The minority carrier recombination lifetimes are tn = 10-6 s
and tp=10-7 s on the p-side and n-side, respectively. Applied bias VA is (kT/q)*ln(108)  0.48V.
What if the n-side is a short base?
The hole diffusion component of the diode saturation current is
calculated using the short-base diode formula:
I0,p = Aqni2Dp/(Wn’×ND)
ref. slide 10