### I–V

```Semiconductor Device Physics
Lecture 8
PN Junction Diodes: I-V Characteristics
Dr. Gaurav Trivedi,
EEE Department,
IIT Guwahati
Empirical Observations of VBR
 VBR decreases with increasing N,
VBR
1
 0.75
NB
 VBR decreases with decreasing EG.
• VBR : breakdown voltage
Dominant breakdown
mechanism is tunneling
Breakdown Voltage, VBR
Breakdown Mechanism: Avalanching
Breakdown Mechanism: Zener Process
Effect of R–G in Depletion Region
Effect of R–G in Depletion Region
Effect of R–G in Depletion Region
Effect of R–G in Depletion Region
Effect of Series Resistance
Effect of High-Level Injection
High-Level Injection Effect
Summary
Minority-Carrier Charge Storage
Charge Control Approach
Charge Control Approach
 Integrating over the n quasineutral region (after all terms multiplied by Adx),
JP ()



d
1
 qA  pn dx    A  dJ P 
dt  xn
p

J p ( xn )
QP
 

qA  pn dx 
 xn

QP
 Furthermore, in a p+n junction,
A
JP ()

0
dJ P   AJ P ()  AJ P ( xn )  AJ P ( xn )
J P ( xn )
 So:
dQP
QP
 AJ P ( xn ) 
dt
p
0
Charge Control Approach
 In steady state, we can calculate pn junction current in two ways:
 From slopes of Δnp(–xp) and Δpn(xn)
 From steady-state charges QN and QP stored in each “excess minority charge
distribution”
dQP
QP
 AJ P ( xn ) 
0
dt
p
 Therefore,
 Similarly,
AJ P ( xn )  I P ( xn ) 
I N ( xp )  
QN
n
QP
p
Charge Control Approach
 Moreover, in a p+n junction:
J N ( xp )  0
J DIFF  J P ( xn )
dQP
QP
 iDIFF 
dt
p
0
Narrow-Base Diode
 Narrow-base diode: a diode where the width of the quasineutral region on the lightly
doped side of the junction is on the order of or less than one diffusion length.
x
0
0
x xc
xp
xn
xc
n-side contact
Narrow-Base Diode I–V
 We have the following boundary conditions:
pn ( x  0)  pn0 (eqVA
kT
1)
pn ( x  xc )  0
 Then, the solution is of the form:
 x LP
x LP

pn ( x )  Ae
 A2e
1
 Applying the boundary conditions, we have:
pn0 (eqVA
kT
 1)  A1  A2
0  A1e xc
LP
 A2e xc
LP
Narrow-Base Diode I–V
 Solving for A1 and A2, and substituting back:
pn ( x)  pn0 (e
 Note that
qVA / kT
 e( xc  x) LP  e( xc  x) LP
 1) 
xc LP
 xc LP
e

e


 , 0  x  xc

e  e
e  e
sinh( ) 
, cosh( ) 
2
2
 The solution can be written more compactly as
pn ( x)  pn0 (e
qVA kT
sinh ( xc  x) LP 
1)
, 0  x  xc
sinh  xc LP 
Narrow-Base Diode I–V
 With decrease base width, xc’0:
pn ( x)  pn0 (e
qVA kT
pn ( x)  pn0 (eqVA
kT
 ( xc  x) LP 
 1) 

 xc LP 
 x 
 1) 1  
 xc 
• Δpn is a linear function of x due to negligible
thermal R–G in region much shorter than
one diffusion length
•  JP is constant
limsinh( )  
 0
lim cosh( )  1 
 0
2
2
• This approximation can be derived using
Taylor series approximation
Narrow-Base Diode I–V
 Because
JP 
pn ( x)
,
then
qD
P
J P  qDP pn0 (e
x
qVA kT
 1 LP  cosh  ( xc  x) LP  
 1) 

sinh  xc LP 


 Then, for a p+n junction:
I DIFF
DP ni2 qVA
 AJ P ( x  0)  qA
(e
LP ND
IDIFF  I0 (eqVA
kT
kT
cosh( xc LP )
 1)
sinh( xc LP )
1)
DP ni 2 cosh( xc LP )
I 0  qA
LP ND sinh( xc LP )
Narrow-Base Diode I–V
 If xc’ << LP,
2

( xc LP )
1
cosh( xc LP )
LP ( xc LP ) LP
2




( xc LP )
sinh( xc LP )
xc
2
xc
 Resulting
DP ni2  LP 
DP ni2
I 0  qA
   qA
LP N D  xc 
xc N D
limsinh( )  
 0
lim cosh( )  1 
 0
2
2
Increase of reverse bias means
• Increase of reverse current
• Increase of depletion width
• Decrease of quasineutral region xc’xc–xn
Wide-Base Diode
 Rewriting the general solution for carrier excess,
pn ( x)  pn0 (e
qVA kT
sinh ( xc  x) / LP 
1)
sinh  xc / LP 
 For the case of wide-base diode (xc’ >> LP),
pn ( x)  pn0 (eqVA
 pn0 (e
kT
qVA kT
 e( xc  x) LP  e( xc  x) LP 
 1) 

xc LP
 xc LP
e
 e

 e xc / LP e x / LP  e xc / LP e x / Lp
 1) 
xc LP
 xc LP
e

e

pn ( x)  pn0 (eqVA
kT
1)e x LP



Back to ideal
diode solution
Wide-Base Diode
 Rewriting the general solution for diffusion current,
I DIFF
DP ni2 qVA
 qA
(e
LP ND
kT
cosh( xc LP )
 1)
sinh( xc LP )
 For the case of wide-base diode (xc’ >> LP),
I DIFF
DP ni2 qVA
 qA
(e
LP ND
e
lim sinh( ) 
 
2
e
lim cosh( ) 
 
2
kT
 1)
Back to ideal
diode solution
Small-Signal Diode Biasing
 When reversed-biased, a pn junction diode becomes functionally equivalent to a
capacitor, whose capacitance decreases as the reverse bias increases.
 Biasing additional a.c. signal va can be viewed as a small oscillation of the depletion width
Y  G  jC
V0 << VA
RS
C
G
Y
: serial resistance
: capacitance
: conductance
Total pn Junction Capacitance

i
R 1 G
va
C  CJ  CD

CJ  A
Minority
carrier
CD 
s
W
 I DC
kT q
Junction / depletion capacitance,
due to variation of depletion charges
Diffusion capacitance,
due to variation of stored minority charges in
the quasineutral regions
• CJ dominates at low forward biases, reverse biases.
• CD dominates at moderate to high forward biases.
Relation Between CJ and VA
 For asymmetrical step junction,
W 
2 s
Vbi  VA 
qN B
NB : bulk semiconductor doping, NA or
ND as appropriate.
 Therefore,
1
W2
2
 2 2
(Vbi  VA )
2
2
CJ
A s
qNB S A
```