6.720J/3.43J - Integrated Microelectronic Devices - Fall 2004
Lecture 14-1
Lecture 14 - p-n Junction (cont.)
March 8, 2007
Contents:
1. Ideal p-n junction out of equilibrium (cont.)
Reading assignment:
del Alamo, Ch. 7, §7.2 (§7.2.3)
Announcements:
Quiz 1: March 13, 7:30-9:30 PM; lec tures #1-12
(up to SCR-type transport). Open book.
Calculator required.
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6.720J/3.43J - Integrated Microelectronic Devices - Fall 2004
Lecture 14-2
Key questions
• What are the dominant physics of current flow in a p-n junction
under bias?
• What underlies the rectifying behavior of the p-n junction?
• What are the key assumptions that allow the development of a
simple model for the I-V characteristics of a p-n diode?
• What are the key dependencies of the current-voltage character­
istics of the p-n diode?
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6.720J/3.43J - Integrated Microelectronic Devices - Fall 2004
Lecture 14-3
1. Ideal p-n junction out of equilibrium (cont.)
� I-V characteristics
Fe=0
Ec
R
G
a) equilibrium
R
G
Ev
Fh=0
Fe
Ec
R
G
b) forward bias
R
G
Ev
Fh
Fe
Ec
R
G
c) reverse bias
R
Fh
G
Ev
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6.720J/3.43J - Integrated Microelectronic Devices - Fall 2004
Lecture 14-4
� In TE:
• balance between electron and hole flows across SCR
• balance between G and R in QNR’s
•I=0
� In forward bias:
• energy barrier to minority carriers reduced
• minority carrier injection
• R > G in QNR’s
• I ∼ eqV/kT
� In reverse bias:
• energy barrier to minority carriers increased
• minority carrier extraction
• G > R in QNR’s
• I saturates to a small value
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6.720J/3.43J - Integrated Microelectronic Devices - Fall 2004
Lecture 14-5
Key thinking to construct model for I-V characteristics:
• junction voltage sets concentration of carriers with enough en­
ergy to get injected;
• rate of carrier injection set by minority carrier transport and
G/R rates in quasi-neutral regions;
• SCR is in quasi-equilibrium.
Fe
Ec
R
G
R
G
Ev
Fh
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6.720J/3.43J - Integrated Microelectronic Devices - Fall 2004
Lecture 14-6
Fe
Ec
R
G
R
G
Ev
Fh
-xp xn
x
Strategy for deriving first-order model for I-V characteristics:
• Compute diode current density as follows:
J = Je(−xp ) + Jh(xn )
• Compute each minority carrier current contribution as follows:
Je (−xp ) = −qn�(−xp)ve (−xp)
Jh(xn ) = qp�(xn )vh(xn )
• Use expressions of ve(−xp ) and vh(xn) derived for similar minor­
ity carrier type problems in Ch. 5.
• Derive expressions for n�(−xp ) and p� (xn) assuming quasi-equilibrium
across the space-charge region.
• Unified result for forward and reverse bias.
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6.720J/3.43J - Integrated Microelectronic Devices - Fall 2004
Lecture 14-7
� Boundary conditions across SCR.
In thermal equilibrium, Boltzmann relations:
φ(xn) − φ(−xp) = φbi =
kT
no (xn)
ln
q
no(−xp )
φ(xn) − φ(−xp) = φbi = −
kT
po (xn)
ln
q
po (−xp )
If net current inside SCR is much smaller than drift and diffusion
components, then quasi-equilibrium ⇒ Boltzmann relations apply:
φbi − V �
kT
n(xn)
ln
q
n(−xp )
φbi − V � −
kT
p(xn )
ln
q
p(−xp )
In LLI, n(xn ) � ND , and p(−xp ) � NA . Also φbi =
Then:
kT
q
A
ln NDnN
2 .
i
n2i
qV
n(−xp) �
exp
NA
kT
2
n
qV
p(xn) � i exp
ND
kT
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6.720J/3.43J - Integrated Microelectronic Devices - Fall 2004
Lecture 14-8
In terms of excesses:
n2i
qV
n (−xp) �
(exp
− 1)
NA
kT
n2i
qV
�
p (xn) �
(exp
− 1)
ND
kT
�
Boundary conditions have all expected features. For electrons (for
example):
• For V = 0:
n� (−xp ) = 0
• For V �
kT
:
q
n2i
qV
n (−xp) �
exp
NA
kT
�
• For V � − kT
:
q
n2i
n (−xp ) � −
NA
�
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6.720J/3.43J - Integrated Microelectronic Devices - Fall 2004
Lecture 14-9
� Minority carrier velocity at edges of SCR (”long” diode: Wn �
Lh, Wp � Le).
Problem identical to ”long bar” studied in Ch. 5: exponentially
decaying excess minority carrier profiles.
n', p'
p'(xn)
n'(-xp)
p'(x)
n'(x)
-wp
-xp
0
x + xp
Le
−x + xn
p� (x) = p� (xn ) exp
Lh
n� (x) = n� (−xp ) exp
wn x
xn
in p-QNR: x ≤ −xp
in n-QNR: x ≥ xn
Carrier velocities:
De
Le
Dh
vh(xn) = vhdiff (xn) =
Lh
ve(−xp ) = vediff (−xp ) = −
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6.720J/3.43J - Integrated Microelectronic Devices - Fall 2004
Lecture 14-10
� Excess minority carrier currents:
Je (−xp ) �
Jh(xn ) �
−qvedif f (−xp) n� (−xp )
qvhdif f (xn) p� (xn)
De n2i
qV
=q
(exp
− 1)
Le NA
kT
Dh n2i
qV
=q
(exp
− 1)
Lh ND
kT
� Total current: sum of electron and hole current:
J = Je(−xp ) + Jh(xn) = qn2i (
1 Dh
qV
1 De
+
)(exp
− 1)
NA Le ND Lh
kT
Define Js ≡saturation current density (A/cm2 ):
J = Js(exp
qV
− 1)
kT
If diode area is A, current is:
I = Is(exp
qV
− 1)
kT
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6.720J/3.43J - Integrated Microelectronic Devices - Fall 2004
I = Is(exp
Lecture 14-11
qV
− 1)
kT
Classic rectifying behavior
I
log |I|
0.43 q
kT
IS
0
0
IS
linear scale
V
0
V
semilogarithmic scale
Should test quality of quasi-equilibrium approximation [problem #7.13].
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6.720J/3.43J - Integrated Microelectronic Devices - Fall 2004
Lecture 14-12
Rectifying behavior arises from boundary conditions across SCR:
n, p
p(x)
n(x)
increasing �
forward bias
increasing �
forward bias
ni2
NA
ni2
ND
increasing�
reverse bias
0
-xp 0
increasing�
reverse bias
xn
x
-In forward bias: carrier concentrations at SCR edges grow up expo­
nentially → I ∼ eqV/kT
-In reverse bias: carrier concentrations at SCR edges reduced quickly
to zero (can’t go below!) → I saturates
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007.
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6.720J/3.43J - Integrated Microelectronic Devices - Fall 2004
Lecture 14-13
Experimental verification:
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6.720J/3.43J - Integrated Microelectronic Devices - Fall 2004
Lecture 14-14
Universality of exponential relationship:
I = IS (exp
qV
− 1)
kT
Then:
I
qV
= exp
−1
IS
kT
Reprinted with permission from Anonymous, G. F. Cerofolini, and M. L. Polignano. "Current-voltage Characteristics of Ideal Silicon
Diodes in the Range 300Ð400 K." Journal of Applied Physics 57, no. 2 (1985): 646-647. Copyright 1985, American Institute of Physics.
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6.720J/3.43J - Integrated Microelectronic Devices - Fall 2004
Lecture 14-15
In short diodes with S = ∞, G&R takes place at surfaces:
n', p'
p'(xn)
n'(-xp)
n'(x)
-wp
p'(x)
-xp
0
xn
wn x
De
wp − xp
Dh
vhdiff (xn ) =
wn − xn
vedif f (−xp) = −
Js is:
Js � qn2i (
1
De
1
Dh
+
)
NA wp − xp ND wn − xn
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6.720J/3.43J - Integrated Microelectronic Devices - Fall 2004
Lecture 14-16
Quasi-Fermi levels across long diode:
Ec
Efe
qV
Efh
Ev
forward bias
Ec
Efh
qV
Efe
Ev
reverse bias
Inside SCR:
Efe − Efh = qV
Then:
np = n2i exp
qV
kT
From here can get also BC’s.
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6.720J/3.43J - Integrated Microelectronic Devices - Fall 2004
Lecture 14-17
Key conclusions
• Forward bias: junction barrier ↓ ⇒ carrier injection ⇒ recom­
bination in QNR’s ⇒ I ∼ eqV/kT
• Reverse bias: junction barrier ↑ ⇒ carrier extraction ⇒ gener­
ation in QNR’s ⇒ I saturates with reverse V
• Rectifying characteristics of pn diode arise from boundary con­
ditions at edges of SCR.
• Excess minority carrier concentration at edges of depletion re­
gion:
n2i
qV
n2i
qV
�
n (−xp ) =
(exp
− 1), p (xn) =
(exp
− 1)
NA
kT
ND
kT
�
• I-V characteristics of ideal pn diode:
I = Is(exp
qV
− 1)
kT
• Quasi-Fermi levels flat across SCR:
np = n2i exp
qV
kT
inside SCR
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