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:
Pnn
x nP
LxpqA
dxxpqAQn
)(
)(
Npp
x pN
LxnqA
dxxnqAQp
)(
)(
EE130/230M Spring 2013 Lecture 13, Slide 2
Derivation of Charge Control ModelConsider the n quasi-neutral region of a forward-biased pn junction:
•The minority carrier diffusion equation is (assuming GL=0):
•Since the electric field is very small,
•Therefore
p
nnP
n p
x
pD
t
p
2
2
xp
PPnqDJ
p
nPn pq
x
J
t
pq
)(
EE130/230M Spring 2013 Lecture 13, Slide 3
• Integrating over the n quasi-neutral region:
• Note that
• So
n
P
npn x
np
J
xJ
P
x
n dxpqAdJAdxpqAt
1)(
)(
)()()()()(
)(
nPnPnPP
J
xJ
P xIxAJxAJAJdJAP
np
p
PnP
P QxI
dt
dQ
)(
EE130/230M Spring 2013 Lecture 13, Slide 4
Derivation Assuming a Long Base
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 excess-minority-charge distribution:
p
PnP
QxI
τ)(
n
NpN
QxI
τ)( Similarly,
0τ
)( p
PnP
P QxI
dt
dQ
EE130/230M Spring 2013 Lecture 13, Slide 5
Charge Control Model
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 quasi-
neutral region
– For holes in narrow n-side:
– Similarly, for electrons in narrow p-side:
P
N
P
Pptr
N
nnP
nPPP
Nnn
W
x nP
D
W
I
Q
W
xpqAD
dx
pdqADAJI
WxpqAdxxpqAQN
n
2 τ
)(
)(2
1)(
2
,
N
Pntr D
W
2τ
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:
– Narrow base:
PkTqV
D
iP Le
N
nqAQ A 1/
2
NkTqV
A
iN Le
N
nqAQ A 1/
2
PkTqV
A
iN We
N
nqAQ A 1
2
1 /2
NkTqV
D
iP We
N
nqAQ A 1
2
1 /2
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):
– Narrow base (both sides):
where and
p
P
n
N QQI
ττ
ptr
P
ntr
N QQI
,, ττ
P
P
p
P
N
N
n
N
L
DL
L
DL
τ and
τNote that
N
Pntr D
W
2τ
2
,
P
Nptr D
W
2τ
2
,
EE130/230M Spring 2013 Lecture 13, Slide 8
Small-Signal Model of the Diode
kTqVkTqV eIdV
deI
dV
d
dV
dI
R/
0A
/0
AA
AA )1(1
qkT
IeI
kT
q
RG DCkTqV
/
1 /0
A
dt
dvC
R
vi aa
EE130/230M Spring 2013 Lecture 13, Slide 9
i
v
+
Small-signalconductance:
Charge Storage in pn Junction Diode
EE130/230M Spring 2013 Lecture 13, Slide 10
pn Junction Small-Signal Capacitance2 types of capacitance associated with a pn junction:
depletion capacitance due to variation of depletion charge
diffusion capacitance–due to variation of stored minority charge in the quasi-neutral regions
For a one-sided p+n junction Q = QP + QN QP so
qkT
IG
dV
dI
dV
dQC P
/
τττ DCp
pA
pA
D
EE130/230M Spring 2013 Lecture 13, Slide 11
A
dep
dV
dQC J
AdV
dQC D
Depletion Capacitance
What are three ways to reduce CJ?
WA
dV
dQC s
A
depJ
EE130/230M Spring 2013 Lecture 13, Slide 12
Total pn-Junction Capacitance
C = CD + CJ
•CD dominates at moderate to high forward biases
•CJ dominates at low forward biases, reverse biases
1/
τ /DCD kTqVAe
qkT
IC
WAC s
J
EE130/230M Spring 2013 Lecture 13, Slide 13
Using C-V Data to Determine Doping
NqA
VV
A
W
C Ss 2Abi
22
2
2J
)(21
EE130/230M Spring 2013 Lecture 13, Slide 14
ExampleIf 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 m2, find the dopant concentration Nl on the more lightly doped side and the dopant concentration Nh on the more heavily doped side. Solution:
ln 2i
lhbi
n
NN
q
kTV 318026.0
84.0
15
202
cm 108.1106
10
eeN
nN kT
qV
l
ih
bi
315
28121923
2
cm 106
)1010106.1102/(2
)/(2
AqslopeN sl
EE130/230M Spring 2013 Lecture 13, Slide 15
Small-Signal Model Summary
qkT
IG DC
/
qkT
IC
/
τ DCD
W
AC s
J
)1( /0
A kTqVDC eII
DJ CCC
Depletion capacitance
Diffusion capacitance
Conductance
EE130/230M Spring 2013 Lecture 13, Slide 16
Transient Response of pn Diode
EE130/230M Spring 2013 Lecture 13, Slide 17
• 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.
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 ChargeConsider a p+n diode (Qp >> Qn):
t
i(t)
t
vA(t)
ts
ts0 pxx
n
qAD
i
dx
dp
n
For t > 0:
pn(x)
EE130/230M Spring 2013 Lecture 13, Slide 19
Storage Delay Time, ts
• ts is the primary “figure of merit” used to characterize the transient response of pn junction diodes
• By separation of variables and integration from t = 0+ to t = ts, noting that
and making the approximation
We conclude that
sp
pR
p
pp ttQ
IQ
idt
dQ
0
ττ
ppF tQI τ/)0(
0)( sp ttQ
R
Fps I
It 1lnτ
EE130/230M Spring 2013 Lecture 13, Slide 20
Qualitative Examples
t
i(t)
ts
Increase IF
t
i(t)
ts
Increase IR
t
i(t)
ts
Decrease p
EE130/230M Spring 2013 Lecture 13, Slide 21
Illustrate how the turn-off transient response would change: