Lecture 15
The pn Junction Diode (II)
Outline
• I-V characteristics
– Forward Bias
– Reverse Bias
Reading Assignment:
Howe and Sodini; Chapter 6, Sections 6.4 - 6.5
6.012 Spring 2007
Lecture 15
1
1. I-V Characteristics (contd.)
Diode Current equation:
⎛ ⎡⎣⎢VVth ⎤⎦⎥ ⎞
I = I o ⎜ e − 1⎟
⎝
⎠
Physics of forward bias:
Fn
p
n
Fp
•
Potential drop across SCR reduced by V
– ⇒ minority carrier injection in QNRs
•
•
•
Minority carrier diffusion through QNRs
Minority carrier recombination at contacts to the QNRs
(and surfaces)
Large supply of carriers injected into the QNRs
⇒
6.012 Spring 2007
[]
I∝e
V
Vth
Lecture 15
2
Fn
p
n
Fp
Physics of reverse bias:
•
Potential drop across SCR increased by V
– ⇒ 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
– ⇒ I saturates to small value
6.012 Spring 2007
Lecture 15
3
Development of analytical current model:
1. Calculate the concentration of minority carriers at
edges of SCR;
2. Find the spatial distribution of the minority carrier
concentrations in each QNR;
3. Calculate minority carrier diffusion current at SCR
edge.
4. Sum minority carrier electron and hole diffusion
currents at SCR edge.
(p-type)
pn(x)
np (x)
(n-type)
contact to
p region
contact to\
n region
n p(– x p) = n po ⋅ e
np(-Wp)=npo
- Wp
VD ⁄ Vth
p n(x n) = p no ⋅ e
<< pno
<< npo
-xp
xn
VD ⁄ Vth
p n(W n) = p no
Wn
x
Reverse Bias
6.012 Spring 2007
Lecture 15
4
Total and Excess Concentrations
Forward Bias (Step 1)
Total Carrier Concentration (n & p)
np(x)
pn(x)
(contact)
(contact)
(p-type)
VD
n po e
(n-type)
p n(x n)
VD
Vth
pnoe
Vth
n p(– xp )
-xp
- Wp
xn
np (–W p) = np o
Wn
x
p n(W n) = p no
Excess Carrier Concentration (n’ & p’) - Subtract npo and pno
n’ p(x)
p’n(x)
(contact)
(contact)
(p-type)
VD
n po (e
(n-type)
p′ n(x n)
Vth
VD
− 1)
pno (e
Vth
−1)
n′ p(–x p)
-xp
- Wp
n′ p(– W p) = 0
6.012 Spring 2007
xn
Wn
x
p′ n(W n) = 0
Lecture 15
5
Steady-State Diffusion
Ink Diffusion Example
•
•
•
•
•
Flux is number of ink molecules passing a plane/cm2-sec
No molecules vanish in the water (NO recombination)
Ink concentration is a constant at x = 0
Ink concentration is zero at x = W (ohmic contact)
Result - Ink concentration falls linearly from x=0 to x=W
fill tube
“ink vacuum”
water
ink
W x
0
Injects
minority
carriers
(a)
fill tube
water
“ink vacuum”
ink
Carriers
recombine at
ohmic contact
W x
0
(b)
nI (x)
nI*
0
W
(c)
6.012 Spring 2007
Lecture 15
6
Minority Carrier Spatial Distribution
(Step 2)
•
Concentration linearly decreases from SCR edge to
ohmic contact. The expressions assumes no
recombination in the QNR.
⎛ n' p (−xp ) ⎞
⎟ • x + xp
n' p (x) = n' p (−xp ) + ⎜⎜
⎟
−
x
W
⎝ p
p ⎠
(
)
⎛ p' n (xn ) ⎞
⎟⎟ • (x − xn )
p' n (x) = p' n (xn ) − ⎜⎜
⎝ Wn − xn ⎠
n’ p (x)
p’ n (x)
(contact)
(contact)
(p-type)
(n-type)
p ′ n( x n)
n ′ p( – x p)
-W p
-x p
n ′ p( – W p) = 0
xn
p ′ n( W n) = 0
Wn
x
Steady-state---> minority carriers are continuously
injected across the junction to maintain the value at the
SCR edge set by the applied bias. The same number
continuously recombine at ohmic contact.
6.012 Spring 2007
Lecture 15
7
Minority Carrier Diffusion Current at
SCR Edge (Step 3)
•
•
•
•
Gradient in minority carrier concentrations across
the n & p QNRs
n = no + n’ ---> dn/dx = dn’/dx
Transport occurs by diffusion
Ignore drift component since minority carriers
At -xp electron diffusion current:
⎛ n' p (−x p ) − 0 ⎞
⎟
Jn = qDn
= qDn ⎜⎜
⎟
dx
−
x
W
⎝
p
p ⎠
dn' p
[]
V
⎡
⎤
⎢ n po (e Vth −1)⎥
⎥
Jn = qDn ⎢
⎢ Wp − x p ⎥
⎢⎣
⎦⎥
⎡
2
ni
Dn
Jn = q
•
• ⎢e
Na Wp − x p ⎢⎣
6.012 Spring 2007
( )− 1⎥⎤
Lecture 15
V
Vth
⎥
⎦
8
Sum minority carrier diffusion
currents at SCR edge (Step 4)
Hole diffusion current at xn by same reasoning:
⎡ (qV ) ⎤
Dp
ni2
Jp = q
•
• ⎢e kT − 1⎥
Nd Wn − xn ⎣⎢
⎥⎦
⎛
Dp ⎞ ⎡
Dn
1
2⎜ 1
⎟ • ⎢e
J = J n + J p = qni ⎜
•
+
•
⎟
⎝ Na Wp − x p Nd Wn − xn ⎠ ⎢⎣
( ) −1⎤⎥
qV
kT
⎥⎦
Current is:
⎛
Dp ⎞ ⎡
Dn
1
2⎜ 1
⎟ • ⎢e
I = qAni ⎜
•
+
•
⎟
⎝ Na Wp − x p N d Wn − x n ⎠ ⎣⎢
( ) − 1⎤⎥
Often written as:
6.012 Spring 2007
qV
kT
⎦⎥
⎡ ⎛ qV ⎞
⎤
⎥
⎢ ⎜⎝ kT ⎟⎠
I = Io ⎢e
− 1⎥
⎢⎣
⎦⎥
Lecture 15
9
Picture of Total Diode Current
Forward Bias
•
•
•
Minority carriers are injected from the other side of the junction
Minority carriers diffuse from SCR edge to the ohmic contact
with no recombination and recombine at contact
Total current found by summing the minority carrier diffusion
currents at SCR edges and assuming no recombination in SCR
n’ p (x)
p’ n (x)
(contact)
(contact)
(p-type)
(n-type)
p ′ n( x n)
n ′ p( – x p)
-W p
-x p
xn
n ′ p( – W p) = 0
p ′ n( W n) = 0
Wn
x
•Majority carriers are transported to the junction from the ohmic
contact by drift and diffusion.
−Wp
6.012 Spring 2007
;;;;;;;;;
;
;
;
;
;;;;;
;
;
;
;
metal
contact to
p region
J
(p-type)
(n-type)
J (total current density)
J (total current density)
Jp (majority holes)
Jp (minority holes)
Jn (minority electrons)
Jn (majority electrons)
0
−xp
0
xn
metal
contact to
n region
Wn
Lecture 15
x
10
Minority Carrier Spatial Distribution
( Reverse Bias)
•
•
•
•
Diode current derivation same for forward and
reverse bias. (same equations for spatial distribution)
Minority carrier concentration at SCR is near zero
under reverse bias.
Concentration linearly increases from SCR edge to
ohmic contact.
Minority carriers flow from contacts to SCR and are
swept across the junction.
(p-type)
pn(x)
np (x)
(n-type)
contact to
p region
contact to\
n region
n p(– x p) = n po ⋅ e
np(-Wp)=npo
- Wp
VD ⁄ Vth
p n(x n) = p no ⋅ e
<< pno
<< npo
-xp
xn
VD ⁄ Vth
p n(W n) = p no
Wn
x
Steady-state---> minority carriers are continuously
extracted across the junction to maintain the value at the
SCR edge set by the applied bias. The same number
continuously are generated at ohmic contact.
6.012 Spring 2007
Lecture 15
11
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
Io
linear scale
6.012 Spring 2007
V
0
V
semilogarithmic scale
Lecture 15
12
What did we learn today?
Summary of Key Concepts
• Diode Current can be analytically
determined by summing the minority
carrier current at both sides of SCR
I = Io
[
]
(e
− 1)
qV
kT
• Under forward bias:
– Minority carriers are injected across the
junction and diffuse to the contact where they
recombine
• Under reverse bias:
– Minority carriers are generated at the contact
and diffuse to the junction where they are
extracted across the junction
6.012 Spring 2007
Lecture 15
13