Lecture 16
The pn Junction Diode (III)
Outline
• I­V Characteristics (Review)
• Small­signal equivalent circuit model
• Carrier charge storage
–Diffusion capacitance
Reading Assignment:
Howe and Sodini; Chapter 6, Sections 6.4 ­ 6.5
6.012 Spring 2009
Lecture 16
1
1. I­V Characteristics (Review)
Diode Current Equation:
 (
I D = I o e

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Lecture 16
VD
Vth
) −1


2
Physics of forward bias:
Diode Current equation:
  qVD  
I D = I o exp
 −1
  kT  
• Junction potential φJ (potential drop across SCR)
reduced by |VD|
– ⇒ minority carrier injection into QNRs
• Minority carrier diffusion through QNRs
• Minority carrier recombination at contacts to the
QNRs (and surfaces)
• Large supply of carriers injected into the QNRs
 qV 
– ⇒ I D ∝ exp  D 
 kT 
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Physics of reverse bias:
  qVD  
I D = I o exp
 −1
  kT  
• Junction potential φJ (potential drop across SCR)
increased by |VD|
– ⇒ minority carrier extraction from QNRs
• Minority carrier diffusion through QNRs
• Minority carrier generation at surfaces & contacts
of QNRs
• Very small supply of carriers available for
extraction
– ⇒ ID saturates to small value
– ⇒
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I D ≈ −I o
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2. Small­signal equivalent circuit model
Examine effect of small signal overlapping bias:
  q(VD + v d )  
i D = I D + i d = I o exp
 − 1
kT
 
 
If vd small enough, linearize exponential characteristics:

 qV
I D + id = I o exp  D
 kT

 qvd

exp



 kT
 
 − 1
 

 qV   qv
= I o exp  D  1 + d
kT
 kT  

 
 − 1
 
  qV  
 qV  qv
= I o exp D  − 1 + I o exp D  d
 kT  kT
  kT  
Then:
id =
q(I D + I o )
• vd
kT
From a small signal point of view. Diode behaves as
conductance of value:
q ( I D + I o ) qI D
gd =
≈
kT
kT
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Small­signal equivalent circuit model
gd
gd depends on bias. In forward bias:
gd =
qI D
kT
gd is linear in diode current.
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Capacitance associated with depletion region:
ρ(x)
+qNd
p-side
(a)
n-side
−xp
x
xn
vD = V D
−qNa
QJ = −qNaxp
ρ(x)
+qNd
p-side −x −x
p
p
(b)
n-side
x
xn x n
vD = VD + vd > VD-->�
xp < xp,| qJ | < | QJ |
−qNa
qJ = −qNaxp
qj = qNaΔxp
Δρ(x) = ρ(x) − ρ(x)
+ qNd
Xd
p-side
(c)
xn xn
n-side
−xp −xp
qj = qj − Qj > 0�
= −qNaxp − (−qNaxp)�
= qNa (xp −xp) �
= qNaΔxp
x
− qNa
−qj = −qNd Δxn
Depletion or junction capacitance:
C j = C j (VD ) =
dqJ
dv D
VD
qεsN a N d
Cj = A
2(N a + N d )(φ B − VD )
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Small­signal equivalent circuit model
gd
Cj
can rewrite as:
qεsN 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
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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? ⇒ quasi­neutrality is
violated.
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excess minoriv carrier concentplation
= excess rnajoriw carrier concentration
A carrier concentrations
In n-type Si, at every x:
In p-type Si, at every x:
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Lecture 16
Quasi­neutrality demands that at every point in QNR:
excess minority carrier concentration
= excess majority carrier concentration
Mathematically:
p′n (x) = p n (x)− p no ≈ n′n (x) = n n (x) − n no
Define integrated carrier charge:
1
p′(xn ) • (Wn − xn )
2
Wn − xn ni2

 qV
= qA
exp  D − 1 = −qNn
2
Nd
 kT

qPn = qA
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Now examine small increase in VD:
n-QNR
n
n(xn)
ΔqNn=-ΔqPn
n(x)
Nd
p
p(xn)
+
ΔqPn
p(x)
ni2
Nd
0
xn
Wn
x
Small increase in VD ⇒ small increase in qPn ⇒ small
increase in |qNn |
Behaves as capacitor of capacitance:
Cdn
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dq
= Pn
dv D
v D = VD
Wn − x n n i2 q
 qV 
= qA
exp  D 
2
N d kT
 kT 
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12
Can write in terms of IDp (portion of diode current due
to holes in n­QNR):
Dp
n i2
q (Wn − x n )
 qV 
qA
exp  D 
=
N d Wn − x n
kT
2Dp
 kT 
2
Cdn
q (Wn − x n )
≈
I Dp
kT
2Dp
2
Define transit time of holes through n­QNR:
τ Tp =
(Wn − x n )2
2D p
Transit time is the average time for a hole to diffuse
through n­QNR [will discuss in more detail in BJT]
Then:
Cdn ≈
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q
• τ Tp • I Dp
kT
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Similarly for p­QNR:
Cdp
q
≈
• τ Tn • I Dn
kT
where τTn is transit time of electrons through p­QNR:
(
Wp − x p
τ Tn =
2D n
)
2
Both capacitors sit in parallel ⇒ total diffusion capacitance:
Cd = Cdn + Cdp =
q
(τTn I Dn + τTp I Dp )
kT
Complete small­signal equivalent circuit model for diode:
gd
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Cj
Cd
14
Bias dependence of Cj and Cd:
C
Cd
C
0
0
Cj
V
• Cj dominates in reverse bias and small forward bias
1
∝
φ B − VD
• Cd dominates in strong forward bias
 qV 
∝ exp  D 
 kT 
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What did we learn today?
Summary of Key Concepts
Large and Small­signal behavior of diode:
• Diode Current:
 qVD 

I = I o  e  kT  − 1


• Conductance: associated with current­voltage
characteristics
– gd ∝ I in forward bias,
– gd negligible in reverse bias
• Junction capacitance: associated with charge
modulation in depletion region
Cj ∝
1
φB − VD
• Diffusion capacitance: associated with charge
storage in QNRs to maintain quasi­neutrality.
Cd ∝ e
6.012 Spring 2009
Lecture 16
 qVD 
 kT 


16
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6.012 Microelectronic Devices and Circuits
Spring 2009
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