6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 16-1
Lecture 16 - The pn Junction Diode (II)
Equivalent Circuit Model
November 3, 2005
Contents:
1. I-V characteristics (cont.)
2. Small-signal equivalent circuit model
3. Carrier charge storage: diffusion capacitance
Reading assignment:
Howe and Sodini, Ch. 6, §§6.4, 6.5, 6.9
Announcements:
Quiz 2: 11/16, 7:30-9:30 PM,
open book, must bring calculator; lectures #10-18.
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 16-2
Key questions
• How does a pn diode look like from a small-signal
point of view?
• What are the leading dependences of the small-signal
elements?
• In addition to the junction capacitance, are there any
other capacitive effects in a pn diode?
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 16-3
1. I-V characteristics (cont.)
Diode current equation:
I = Io(exp
qV
− 1)
kT
Physics of forward bias:
Fp
p
n
Fn
• potential difference across SCR reduced by V ⇒ minority carrier injection in QNR’s
• minority carrier diffusion through QNR’s
• minority carrier recombination at surface of QNR’s
• large supply of carriers available for injection
⇒ I ∝ eqV /kT
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 16-4
Fp
p
n
Fn
Physics of reverse bias:
• potential difference across SCR increased by V
⇒ minority carrier extraction from QNR’s
• minority carrier diffusion through QNR’s
• minority carrier generation at surface of QNR’s
• very small supply of carriers available for extraction
⇒ I saturates to small value
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 16-5
I-V characteristics: I = Io(exp qV
− 1)
kT
I
log |I|
0.43 q
kT
=60 mV/dec @ 300K
Io
0
0
Io
linear scale
V
0
semilogarithmic scale
V
6.012 - Microelectronic Devices and Circuits - Fall 2005
Source/drain-body pn diode of NMOSFET:
Lecture 16-6
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 16-7
Key dependences of diode current:
I = qAn2i (
Dn
Dp
1
1
qV
− 1)
+
)(exp
Na Wp − xp Nd Wn − xn
kT
n2i
qV
(exp
N
kT
•I∝
− 1) ≡ excess minority carrier concentration at edges of SCR
n2i
N
– in forward bias: I ∝ exp qV
kT : the more carrier
are injected, the more current flows
n2i
−N :
– in reverse bias: I ∝
the minority carrier
concentration drops to negligible values and the
current saturates
• I ∝ D: faster diffusion ⇒ more current
1
: shorter region to diffuse through ⇒ more
• I ∝ WQNR
current
• I ∝ A: bigger diode ⇒ more current
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 16-8
2. Small-signal equivalent circuit model
Examine effect of small signal overlapping bias:
q(V + v)
− 1]
I + i = Io[exp
kT
If v small enough, linearize exponential characteristics:
I + i = Io(exp
qv
qv
qV
qV
exp
− 1) ' Io [exp
(1 + ) − 1]
kT
kT
kT
kT
= Io(exp
qV
qV qv
− 1) + Io(exp
)
kT
kT kT
Then:
q(I + Io)
i=
v
kT
From small signal point of view, diode behaves as conductance of value:
gd =
q(I + Io)
kT
6.012 - Microelectronic Devices and Circuits - Fall 2005
Small-signal equivalent circuit model, so far:
gd
gd depends on bias. In forward bias:
qI
gd '
kT
gd is linear in diode current.
Lecture 16-9
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 16-10
Must add capacitance associated with depletion region:
gd
Cj
Depletion or junction capacitance:
Cj =
Cjo
1 − φVB
s
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 16-11
3. Carrier charge storage: diffusion capacitance
What happens to the majority carriers?
Carrier picture so far:
log p, n
Na
po
Nd
no
p
n
ni2
Nd
ni2
Na
0
x
If in QNR minority carrier concentration ↑ but majority
carrier concentration unchanged
⇒ quasi-neutrality is violated.
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 16-12
Quasi-neutrality demands that at every point in QNR:
excess minority carrier concentration
= excess majority carrier concentration
n-QNR
n
n(xn)
n(x)
qNn
Nd
p
p(xn)
p(x)
qPn
ni2
Nd
0
xn
Wn
x
Mathematically:
p0 (x) = p(x) − po ' n0 (x) = n(x) − no
Define integrated carrier charge:
qP n = qA 12 p0 (xn)(Wn − xn) =
=
2
Wn −xn ni
qA 2 Nd (exp qV
kT
− 1) = −qN n
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 16-13
Now examine small increase in V :
n-QNR
n
V
+-
n(xn)
-
I
∆qNn=-∆qPn
p
n(x)
n
Nd
p
p(xn)
+
∆qPn
p(x)
ni2
Nd
0
xn
Wn
x
Small increase in V ⇒ small increase in qP n ⇒ small
increase in |qN n|
Behaves as capacitor of capacitance:
Cdn =
dqP n
|V
dV
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 16-14
Can write qP n in terms of Ip (portion of diode current
due to holes in n-QNR):
qP n
Dp
(Wn − xn)2 n2i
qV
− 1)
=
qA
(exp
2Dp
Nd Wn − xn
kT
(Wn − xn)2
=
Ip
2Dp
Define transit time of holes through n-QNR:
τT p
(Wn − xn)2
=
2Dp
Transit time is average time for a hole to diffuse through
n-QNR [will discuss in more detail in BJT]
Then:
qP n = τT p Ip
and
Cdn '
q
τT pIp
kT
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 16-15
Similarly for p-QNR:
qN p = τT nIn
Cdp
q
'
τT nIn
kT
where τT n is transit time of electrons through p-QNR:
τT n
(Wp − xp)2
=
2Dn
Both capacitors sit in parallel ⇒ total diffusion capacitance:
Cd = Cdn + Cdp =
q
q
(τT nIn + τT pIp ) =
τT I
kT
kT
with:
τT nIn + τT p Ip
τT =
I
Note that
qP n + qN p = τT nIn + τT p Ip = τT I
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 16-16
Complete small-signal equivalent circuit model for diode:
gd
Cj
Cd
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 16-17
Bias dependence of Cj and Cd :
C
Cd
C
2Cjo
0
Cj
φB/2 V
0
• Cd dominates in strong forward bias (∼ eqV /kT )
dominates in reverse bias and small forward bias
• Cj √
(∼ 1/ φB − V )
- For strong forward bias, model for Cj invalid (doesn’t
blow up)
- Common ”hack”, let Cj saturate at value corresponding to V = φ2B
√
Cj,max =
2Cjo
6.012 - Microelectronic Devices and Circuits - Fall 2005
Lecture 16-18
Key conclusions
Small-signal behavior of diode:
• conductance: associated with current-voltage characteristics
gd ∼ I in forward bias, negligible in reverse bias
• junction capacitance: associated with charge modulation in depletion region
r
Cj ∼ 1/ φB − V
• diffusion capacitance: associated with charge storage
in QNR’s to keep quasi-neutrality
Cd ∼ eqV /kT
Cd ∼ I