Lecture 11
OUTLINE
• pn Junction Diodes (cont’d)
– Narrow-base diode
– Junction breakdown
Reading: Pierret 6.3.2, 6.2.2; Hu 4.5
Introduction
• The ideal diode equation was derived assuming that the lengths of the
quasi-neutral p-type & n-type regions (WP’ , WN’) are much greater
than the minority-carrier diffusion lengths (Ln , Lp) in these regions.
 Excess carrier concentrations decay exponentially to 0.
 Minority carrier diffusion currents decay exponentially to 0.
• 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 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.
EE130/230A Fall 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)
EE130/230A Fall 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:
EE130/230A Fall 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
EE130/230A Fall 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
EE130/230A Fall 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.)
EE130/230A Fall 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
EE130/230A Fall 2013

Lecture 11, Slide 8
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 9
pn Junction Breakdown
C. C. Hu, Modern Semiconductor Devices for Integrated Circuits, Figure 4-10
Breakdown
voltage, VBR
VA
A Zener diode is designed
to operate in the
breakdown mode:
EE130/230A Fall 2013
Lecture 11, Slide 10
Review: Peak E-Field in a pn Junction
E(x)



dx
 Si
 (0) 
qN A x p
 Si
-xp

qN D xn
 Si
2qVbi  VA  N A N D

 Si
N A  ND
xn
E(0)
For a one-sided junction,
 (0) 
2qVbi  VA N
 Si
where N is the dopant concentration on the lightly doped side
EE130/230A Fall 2013
Lecture 11, Slide 11
x
Breakdown Voltage, VBR
• If the reverse bias voltage (-VA) is so large that the peak electric
field exceeds a critical value ECR, then the junction will “break
down” (i.e. large reverse current will flow)

CR
2qN Vbi  VBR 

s
• Thus, the reverse bias at which breakdown occurs is
VBR 
EE130/230A Fall 2013
 s CR
2qN
2
 Vbi
Lecture 11, Slide 12
Avalanche Breakdown Mechanism
R. F. Pierret, Semiconductor Device Fundamentals, Figure 6.12
High E-field:
VBR 
 s CR
2qN
2
if VBR >> Vbi
Low E-field:
ECR increases slightly with N:
For 1014 cm-3 < N < 1018 cm-3,
105 V/cm < ECR < 106 V/cm
EE130/230A Fall 2013
Lecture 11, Slide 13
Tunneling (Zener) Breakdown Mechanism
Dominant breakdown mechanism when both sides of a junction
are very heavily doped.
VA = 0
VA < 0
Ec
Ev
VBR 

CR
 s CR
2qN
2
 Vbi
 106 V/cm
Typically, VBR < 5 V for Zener breakdown
EE130/230A Fall 2013
Lecture 11, Slide 14
C. C. Hu, Modern Semiconductor Devices for Integrated Circuits, Figure 4-12
Empirical Observations of VBR
R. F. Pierret, Semiconductor Device Fundamentals, Figure 6.11
• VBR decreases with
increasing N
• VBR decreases with
decreasing EG
EE130/230A Fall 2013
Lecture 11, Slide 15
VBR Temperature Dependence
• For the avalanche mechanism:
– VBR increases with increasing T, because the mean free
path decreases
• For the tunneling mechanism:
– VBR decreases with increasing T, because the flux of
valence-band electrons available for tunneling increases
EE130/230A Fall 2013
Lecture 11, Slide 16
Summary: Junction Breakdown
•
If the peak electric field in the depletion region exceeds a
critical value ECR, then large reverse current will flow.
This occurs at a negative bias voltage called the breakdown
voltage, VBR:
VBR 
 s CR
2qN
2
 Vbi
where N is the dopant concentration on the more lightly doped side
•
The dominant breakdown mechanism is
avalanche, if N < ~1018/cm3
tunneling, if N > ~1018/cm3
EE130/230A Fall 2013
Lecture 11, Slide 17