Lecture 16
The pn Junction Diode (III)
Outline
• Small-signal equivalent circuit model
• Carrier charge storage
–Diffusion capacitance
Reading Assignment:
Howe and Sodini; Chapter 6, Sections 6.4 - 6.5
6.012 Spring 2007
Lecture 16
1
I-V Characteristics
Diode Current equation:
( )
⎡
I = Io ⎢e
⎢
⎣
I
V
Vth
⎤
−1⎥
⎥
⎦
lg |I|
0.43 q
kT
=60 mV/dec @ 300K
Io
0
0
V
Io
linear scale
6.012 Spring 2007
0
V
semilogarithmic scale
Lecture 16
2
2. Small-signal equivalent circuit model
Examine effect of small signal adding to forward bias:
⎛ q( V+v) ⎞
⎡ ⎛ q (V+v )⎞
⎤
⎜
⎟
⎢ ⎜⎝ kT ⎟⎠
⎥
I + i = Io ⎢e
−1⎥ ≈ I oe ⎝ kT ⎠
⎢
⎥
⎣
⎦
If v small enough, linearize exponential characteristics:
⎡ (qV ) (qv )⎤
⎡ (qV )⎛
qv ⎞ ⎤
kT
kT
kT
I + i ≈ I o ⎢e
e ⎥ ≈ I o ⎢e
⎜1 +
⎟⎥
⎝
kT ⎠ ⎥⎦
⎢⎣
⎥⎦
⎢⎣
= Io
Then:
qv
(
)
(
)
e
+I e
qV
kT
o
qV
kT
kT
qI
i=
•v
kT
From a small signal point of view. Diode behaves as
conductance of value:
qI
gd =
kT
6.012 Spring 2007
Lecture 16
3
Small-signal equivalent circuit model
gd
gd depends on bias. In forward bias:
qI
gd =
kT
gd is linear in diode current.
6.012 Spring 2007
Lecture 16
4
Capacitance associated with depletion region:
ρ(x)
p-side
(a)
−xp
+qNd
n-side
x
xn
vD = V D
−qNa
QJ = −qNaxp
ρ(x)
+qNd
p-side −x −x
p
p
(b)
n-side
x
xn xn
vD = VD + vd > VD-->
xp < xp,| qJ | < | QJ |
−qNa
qJ = −qNaxp
qj = qNa∆xp
∆ρ(x) = ρ(x) − ρ(x)
+ qNd
Xd
p-side
(c)
xn xn
−xp −xp
qj = qj − Qj > 0
= −qNaxp − (−qNaxp)
= qNa (xp −xp)
= qNa∆xp
n-side
x
− qNa
−qj = −qNd ∆xn
Depletion or junction capacitance:
dqJ
C j = C j (VD ) =
dv D
Cj = A
6.012 Spring 2007
VD
qε s N a N d
2(N a + N d )(φ B − VD )
Lecture 16
5
Small-signal equivalent circuit model
gd
Cj
can rewrite as:
qε s N a N d
Cj = A
•
2(N a + N d )φ B
or,
Cj =
φB
(φ B − VD )
C jo
1−
Under Forward Bias assume
VD
φB
VD ≈
φB
2
C j = 2C jo
Cjo ≡ zero-voltage junction capacitance
6.012 Spring 2007
Lecture 16
6
3. Charge Carrier Storage:
diffusion capacitance
What happens to majority carriers?
Carrier picture thus far:
ln p, n
Na
Nd
po
no
p
n
ni2
Nd
ni2
Na
0
x
If QNR minority carrier concentration ↑ but majority
carrier concentration unchanged? ⇒ quasi-neutrality is
violated.
6.012 Spring 2007
Lecture 16
7
Quasi-neutrality demands that at every point in QNR:
excess minority carrier concentration
= excess majority carrier concentration
(p-type)
pp(x) = Na + np(x)
nn(x) = Nd + pn(x)
ppo = Na
metal
contact to
p region
pn(xn) = pno . eVD/Vth
pn(x)
np(x)
np(−Wp) = npo
metal
contact to
n region
nno = Nd
np(−xp) = npo . eVD/Vth
−Wp
;;;;;;;;;;;;;;;;;;
;;;;;;;;;;;;;;;;;;
;
;
;
;
;
;
;
;
;
carrier concentrations
(cm−3)
(n-type)
0
−xp
0
xn
pn(Wn) = pno
Wn
x
Mathematically:
pn (x) − pno = n n (x) − n no
Define integrated carrier charge:
1
q Pn = qA (pn (x n ) − pno ) • (W n − x n )
2
W n − x n n i2 ⎡ qVkT ⎤
= qA
e −1⎥ = −q Nn
⎢
⎣
⎦
2
Nd
6.012 Spring 2007
Lecture 16
8
Now examine small increase in V:
;;;;;;;;;;;;;;;;;;
metal
contact to
p region
p-type
n-type
nn(xn)
[for VD + vd]
pp(−xp)
[for VD + vd]
dqPp
pp(−xp)
[for VD]
nn(xn)
[for VD]
;;;;;;;;;;;;;;;;;;;
;
;
;
;
;
;
;
;
;
carrier concentrations (cm−3)
−dqNn
pn(xn)
[for VD + vd]
np(−xp)
[for VD + vd]
pn(xn)
[for VD]
−dqNp
np(−xp)
[for VD]
0
−xp
−Wp
0
xn
dqPn
metal
contact to
n region
Wn
x
Small increase in V ⇒ small increase in qPn ⇒ small
increase in |qNn |
Behaves as capacitor of capacitance:
Cdn =
6.012 Spring 2007
dqPn
dV
= qA
VD
Lecture 16
Wn − xn n q
e
2
N d kT
2
i
⎡ qVD ⎤
⎢
⎥
⎣ kT ⎦
9
Similarly for p-QNR:
Cdp =
Wp − xp n q
= qA
e
2
N a kT
2
i
dqNp
dV
VD
⎡ qVD ⎤
⎢
⎥
⎣ kT ⎦
Both capacitors sit in parallel ⇒ total diffusion capacitance:
C d = Cdn + Cdp
Complete small-signal equivalent circuit model for diode:
gd
6.012 Spring 2007
Lecture 16
Cj
Cd
10
Bias dependence of Cj and Cd:
C
Cd
C
Cj
0
0
V
• Cj dominates in reverse bias and small forward bias
∝
1
φB − V
• Cd dominates in strong forward bias
∝e
6.012 Spring 2007
Lecture 16
⎡ qV ⎤
⎢ ⎥
⎣ kT ⎦
11
What did we learn today?
Summary of Key Concepts
Large and Small-signal behavior of diode:
•
Diode Current:
I = Io
•
[
]
(e
− 1)
qV
kT
Conductance: associated with current-voltage
characteristics
– gd ∝ I in forward bias,
– gd negligible in reverse bias
•
Junction capacitance: associated with charge
modulation in depletion region
Cj ∝
•
1
φB − V
Diffusion capacitance: associated with charge
storage in QNRs to maintain quasi-neutrality.
Cd ∝ e
6.012 Spring 2007
Lecture 16
⎡ qV ⎤
⎢ ⎥
⎣ kT ⎦
12