Download Lecture 1 - Inst.eecs.berkeley.edu

Lecture 13
OUTLINE
• pn Junction Diodes (cont’d)
– Charge control model
– Small-signal model
– Transient response: turn-off
Reading: Pierret 6.3.1, 7, 8.1; Hu 4.4, 4.10-4.11
Minority-Carrier Charge Storage
• Under forward bias (VA > 0), excess minority carriers are
stored in the quasi-neutral regions of a pn junction:
QN  qA

xp
n p ( x)dx

QP  qA pn ( x)dx
xn
 qApn ( xn ) LP
 qAn p ( x p ) LN
EE130/230M Spring 2013
Lecture 13, Slide 2
Derivation of Charge Control Model
Consider the n quasi-neutral region of a forward-biased pn junction:
•The minority carrier diffusion equation is (assuming GL=0):
pn
 2 pn pn
 DP

2
t
x
p
•Since the electric field is very small,
•Therefore
J P  qDP
(qpn )
J P qpn


t
x
p
EE130/230M Spring 2013
Lecture 13, Slide 3
p n
x
Derivation Assuming a Long Base
• Integrating over the n quasi-neutral region:
J P ()






1 
qA  pn dx    A  dJ P  qA  pn dx 
t  x n
 p  xn


J p ( xn )
• Note that  A
J P ()
 dJ
P
  AJ P ()  AJ P ( xn )  AJ P ( xn )  I P ( xn )
J p ( xn )
dQP
QP
 I P ( xn ) 
• So
dt
p
EE130/230M Spring 2013
Lecture 13, Slide 4
Charge Control Model
We can calculate pn-junction current in 2 ways:
1. From slopes of np(-xp) and pn(xn)
2. From steady-state charges QN, QP stored in each excessminority-charge distribution:
dQP
QP
 I P ( xn ) 
0
dt
τp
QP
 I P ( xn ) 
τp
 QN
Similarly, I N ( x p ) 
τn
EE130/230M Spring 2013
Lecture 13, Slide 5
Charge Control Model for Narrow Base
•
For a narrow-base diode, replace p and/or n by
the minority-carrier transit time tr
–
time required for minority carrier to travel across the quasineutral region
– For holes in narrow n-side:
1
QP  qA pn ( x)dx  qA pn ( xn )WN
xn
2
d p n
p n ( x n )
I P  AJ P   qADP
 qADP
dx
WN
WN
QP WN 
 τ tr , p 

IP
2 DP
2

WP 
– Similarly, for electrons in narrow p-side: τ tr ,n 
2 DN
2
EE130/230M Spring 2013
Lecture 13, Slide 6
Charge Control Model Summary
• Under forward bias, minority-carrier charge is stored in the
quasi-neutral regions of a pn diode.
– Long base:


ni2 qVA / kT
QN  qA
e
 1 LN
NA


ni2 qVA / kT
QP  qA
e
 1 LP
ND
– Narrow base:


1 ni2 qVA / kT
QN  qA
e
 1 WP
2 NA


1 ni2 qVA / kT
QP  qA
e
 1 WN
2 ND
EE130/230M Spring 2013
Lecture 13, Slide 7
• The steady-state diode current can be viewed as the
charge supply required to compensate for charge loss
via recombination (for long base) or collection at the
contacts (for narrow base).
– Long base (both sides): I 
 QN QP

τn
τp
 QN QP
– Narrow base (both sides): I 

τtr ,n τtr , p
where
τ tr ,n
2

WP 

2 DN
and τ tr , p
2

WN 

2 DP
Note that
EE130/230M Spring 2013
Lecture 13, Slide 8
LN DN
L
D

and P  P
τn
LN
τ p LP
Small-Signal Model of the Diode
i
+
va
dva
i  C
R
dt
v

1
dI
d
d
qVA / kT


I 0 (e
 1) 
I 0 e qVA / kT
R dVA dVA
dVA
Small-signal
I DC
1
q
qVA / kT
conductance: G 

I 0e

R
EE130/230M Spring 2013
kT
Lecture 13, Slide 9
kT / q
Charge Storage in pn Junction Diode
EE130/230M Spring 2013
Lecture 13, Slide 10
pn Junction Small-Signal Capacitance
2 types of capacitance associated with a pn junction:
depletion capacitance CJ 
 due to variation of depletion charge
diffusion capacitance
dQdep
dVA
dQ
CD 
dVA
–due to variation of stored
minority charge in the quasi-neutral regions
For a one-sided p+n junction Q = QP + QN  QP so
τ p I DC
dQP
dI
CD 
 τp
 τ pG 
dVA
dVA
kT / q
EE130/230M Spring 2013
Lecture 13, Slide 11
Depletion Capacitance
CJ 
dQdep
dVA
A
s
W
What are three ways to reduce CJ?
EE130/230M Spring 2013
Lecture 13, Slide 12
Total pn-Junction Capacitance
C = CD + C J


τI DC
CD 
 e qVA / kT  1
kT / q
CJ  A
s
W
•CD dominates at moderate to high forward biases
•CJ dominates at low forward biases, reverse biases
EE130/230M Spring 2013
Lecture 13, Slide 13
Using C-V Data to Determine Doping
2(Vbi  VA )
1
W2
 2 2  2
2
A q S N
CJ
A s
EE130/230M Spring 2013
Lecture 13, Slide 14
Example
If the slope of the (1/C)2 vs. VA characteristic is -2x1023 F-2 V-1,
the intercept is 0.84V, and A is 1 mm2, find the dopant
concentration Nl on the more lightly doped side and the
dopant concentration Nh on the more heavily doped side.
Solution: N l  2 /( slope  q s A2 )
 2 /( 2 10 1.6 10
23
19
10
12
 10
)
8 2
 6 1015 cm 3
2
qV
0.84
ni kTbi
10 20 0.026
kT N h Nl
18
3
Vbi 
ln

N

e

e

1
.
8

10
cm
h
2
q
Nl
6 1015
ni
EE130/230M Spring 2013
Lecture 13, Slide 15
Small-Signal Model Summary
C  C J  CD
I DC  I 0 (e qVA / kT  1)
A s
Depletion capacitance C J 
W
τI DC
Diffusion capacitance CD 
kT / q
EE130/230M Spring 2013
I DC
Conductance G 
kT / q
Lecture 13, Slide 16
Transient Response of pn Diode
• Suppose a pn-diode is forward biased, then suddenly turned
off at time t = 0. Because of CD, the voltage across the pn
junction depletion region cannot be changed instantaneously.
The delay in switching between
the ON and OFF states is due
to the time required to change
the amount of excess minority
carriers stored in the
quasi-neutral regions.
EE130/230M Spring 2013
Lecture 13, Slide 17
Turn-Off Transient
• In order to turn the diode off, the excess minority
carriers must be removed by net carrier flow out of
the quasi-neutral regions and/or recombination
– Carrier flow is limited by the switching circuitry
EE130/230M Spring 2013
Lecture 13, Slide 18
Decay of Stored Charge
Consider a p+n diode (Qp >> Qn):
pn(x)
i(t)
ts
t
vA(t)
For t > 0:
dpn
dx
EE130/230M Spring 2013
x  xn
i

0
qADp
Lecture 13, Slide 19
ts
t
Storage Delay Time, ts
• ts is the primary “figure of merit” used to characterize the
transient response of pn junction diodes

Qp 

i
  I R 


dt
τp
τ
p


dQ p
Qp
0  t  ts
• By separation of variables and integration from t = 0+ to t = ts,
noting that I F  Q p (t  0) / τ p
and making the approximation Q p (t  t s )  0
 IF 

We conclude that t s  τ p ln 1 
 IR 
EE130/230M Spring 2013
Lecture 13, Slide 20
Qualitative Examples
Illustrate how the turn-off transient response would change:
Increase IF
Decrease p
Increase IR
i(t)
i(t)
ts
EE130/230M Spring 2013
t
i(t)
ts
Lecture 13, Slide 21
t
ts
t