Lecture 11
OUTLINE
• pn Junction Diodes (cont’d)
– Narrow-base diode
Reading: Pierret 6.3.2
Introduction
• The ideal diode equation was derived assuming that the lengths of
the quasi-neutral p-type and n-type regions (WP’ , WN’) are much
greater than the minority-carrier diffusion lengths (Ln , Lp) so that
the excess carrier concentrations decay exponentially to 0 hence
the minority carrier diffusion currents decay exponentially to 0
within these regions, due to recombination.
• In modern IC devices, however, it is common for one side of a pn
junction to be shorter than the minority-carrier diffusion length,
so that a significant fraction of the “injected” minority carriers
reach the end of the quasi-neutral region, at the metal contact.
– Recall from Lecture 8 that Dp = Dn = 0 at an ohmic contact.
 In this lecture we re-derive the diode I-V equation with the
boundary condition that Dp = 0 at a distance xc’ (rather than )
from the edge of the depletion region.
EE130/230M Spring 2013
Lecture 11, Slide 2
Excess Carrier Distribution (n side)
2
d
• From the minority carrier diffusion equation: Dpn  Dpn  Dp2n
dx 2
Dp p Lp
• For convenience, let’s use the coordinate system:
x’’
0
0
x’
xc'
• So the solution is of the form: Dpn ( x' )  A1e
x '/ L p
 A2e
• We have the following boundary conditions:
Dpn ( x'  0)  pno (e qVA / kT  1)
EE130/230M Spring 2013
Lecture 11, Slide 3
Dpn ( x'  xc ' )  0
 x '/ L p
• Applying the boundary conditions, we have:
Dpn (0)  A1  A2  pno (e qVA / kT  1)
• Therefore
Dpn ( x )  A1e
'
c
xc' / L p
 A2 e
 xc' / L p
0
 e xc  x ' / LP  e  xc  x ' / LP
 1)
 e xc' / LP  e  xc' / LP

'
Dpn ( x' )  pn 0 (e
• Since
qVA / kT
sinh   
Dpn ( x' )  pn 0 (e
e e  
2
qVA / kT
'

, 0  x'  xc'


this can be rewritten as


 sinh xc'  x' / LP
 1)
'
sinh
x
c / LP



, 0  x'  x


• We need to take the derivative of Dpn’ to obtain the hole
diffusion current within the quasi-neutral n region:
EE130/230M Spring 2013
Lecture 11, Slide 4
'
c
Dpn ( x)
J P   qD p
x
 1


coshxc  x / LP 

 LP

qVA / kT
J p  qD p pn 0 e
1 

sinh xc / LP 




e e 


where cosh  


2
Evaluate Jp at x=xn (x’=0) to find the injected hole
current:
D n2
cosh x / L 
Jp
x  0
q
p
i
LP N D
(e qVA
kT
 1)
c
P
sinh xc / LP 
Thus, for a one-sided p+n junction (in which the current
is dominated by injection of holes into the n-side) with a
short n-side:
I  I 0 (e
qV A kT
EE130/230M Spring 2013
 1) where I  qA
'
0
Lecture 11, Slide 5


DP ni 2 cosh xc' / LP
LP N D sinh xc' / LP


sinh    
as   0 and cosh   1   2 as   0
Therefore if xc’ << LP:
2


cosh  xc / LP  1   xc / LP 
LP


xc / LP 
sinh  xc / LP 
xc
For a one-sided p+n junction, then:
D p ni2
I 0  qA
LP N D
EE130/230M Spring 2013
D p ni2
 LP 
   qA
xc N D
 xc 
Lecture 11, Slide 6
Excess Hole Concentration Profile
If xc’ << LP:
Dpn ( x' )  pn 0 (e
 p n 0 (e
qVA / kT
qVA / kT


 
  
 sinh xc'  x' / LP
 1)
'
sinh
x
c / LP


 xc'  x' / LP 

x' 
qVA / kT
  pn 0 (e
 1)
 1)1  ' 
'
 xc / LP 
 xc 
Dpn is a linear function:
pno (e qVA / kT
Dpn(x)
 1)
0
 Jp is constant
0
slope is
constant
x'c
x'
(No holes are lost due to recombination as they diffuse to the metal contact.)
EE130/230M Spring 2013
Lecture 11, Slide 7
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
EE130/230M Spring 2013

Lecture 11, Slide 8
xn
x
Summary
• 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
EE130/230M Spring 2013
Lecture 11, Slide 9