6.720J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 10-1
Lecture 10 - Carrier Flow (cont.)
February 28, 2007
Contents:
1. Minority-carrier type situations
Reading assignment:
del Alamo, Ch. 5, §5.6
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 10-2
Key questions
• What characterizes minority-carrier type situations?
• What is the length scale for minority-carrier type situations?
• What do majority carriers do in minority-carrier type situations?
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 10-3
Overview of simplified carrier flow formulations
General drift-diffusion�
situation�
(Shockley's equations)
1D approx.
Quasi-neutral�
situation�
(negligible volume�
charge)
Majority-carrier �
type situation�
(V=0, n'=p'=0)
Space-charge�
situation�
(field independent�
of n, p)
Minority-carrier �
type situation�
(V=0, n'=p'=0, LLI)
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 10-4
Simplified set of Shockley equations
for 1D quasi-neutral situations
p − n + ND − NA � 0
Je = −qnvedrift + qDe ∂n
∂x
∂p
Jh = qpvhdrif t − qDh ∂x
∂n
∂t
e
=
Gext − U
+
1q
∂J
∂x
or
∂Jt
∂x
∂p
∂t
h
= Gext − U −
1q ∂J
∂x
�0
Jt = Je + Jh
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 10-5
1. Minority-carrier type situations
Situations characterized by:
• excess carriers over TE
• no external electric field applied (but small internal field gener­
ated by carrier injection: E = Eo + E � )
Example: electron transport through base of npn BJT.
Two approximations:
1. E small ⇒ |v drift | ∝ |E|
2. Low-level injection
⇒ for n-type:
• n � no
• p � p�
•U�
p�
τ
• negligible minority carrier drift due to E �
(but can’t say the same about majority carriers)
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 10-6
Shockley equations for 1D quasi-neutral situations
p − n + ND − NA � 0
Je = −qnvedrift + qDe ∂n
∂x
∂p
Jh = qpvhdrif t − qDh ∂x
∂n
∂t
e
= Gext − U + 1q ∂J
∂x
∂p
∂t
or
∂Jt
∂x
h
= Gext − U − q1 ∂J
∂x
� 0
Jt = Je + Jh
� Further simplifications for n-type minority-carrier-type situations
• Majority-carrier current equation:
∂no ∂n�
Je � q(no + n )µe (Eo + E ) + qDe (
+
)
∂x
∂x
�
�
but in TE:
Jeo = qnoµe Eo + qDe
∂no
=0
∂x
Then:
∂n�
Je � qno µe E + qn µeEo + qDe
∂x
�
�
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 10-7
• Minority-carrier current equation:
∂po ∂p�
Jh � q(po + p )µh (Eo + E ) − qDh(
+
)
∂x
∂x
�
�
In TE, Jho = 0, and:
∂p�
∂p�
�
Jh � qp µhEo + qp µhE − qDh
� qp µh Eo − qDh
∂x
∂x
�
�
�
• Minority-carrier continuity equation:
∂p�
p� 1 ∂Jh
= Gext − −
τ
q ∂x
∂t
Now plug in Jh from above:
∂ 2p�
∂p� p�
∂p�
Dh 2 − µhEo
− + Gext =
∂x
∂x
τ
∂t
One differential equation with one unknown: p� .
If Gext and BC’s are specified, problem can be solved.
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 10-8
Shockley equations for
1D minority-carrier type situations
n-type
p-type
po − no + ND − NA � 0
p� � n�
�
�
Je = qno µeE � + qn�µe Eo + qDe ∂n
∂x
�
�
Jh = qp� µhEo − qDh ∂p
∂x
2 �
�
�
Je = qn�µe Eo + qDe ∂n
∂x
Jh = qpo µhE � + qp�µh Eo − qDh ∂p
∂x
Dh ∂∂xp2 − µhEo ∂p
− pτ + Gext =
∂x
∂p�
∂t
∂Jt
∂x
2 �
�
�
De ∂∂xn2 + µe Eo ∂n
− nτ + Gext =
∂x
∂n�
∂t
�0
Jt = Je + Jh
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 10-9
Example 1:
Diffusion and bulk recombination in a ”long” bar
Uniform doping: Eo = 0; static conditions:
∂
∂t
=0
hυ
n
gl
x
0
Minority carrier profile:
p'
p'(0)
0
x
Majority carrier profile? n� = p� exactly?
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6.720J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 10-10
p'
p'(0)
0
x
Far away Jt = 0 ⇒ Jt = 0 everywhere.
dp�
Jt = Je + Jh � qnoµe E + q(De − Dh)
=0
dx
�
If De = Dh ⇒ diffusion term = 0
⇒ drift term = 0
⇒ E � = 0
⇒ n� = p�
But, typically De > Dh ⇒ diffusion term < 0 (for x > 0)
⇒ drift term > 0
⇒ E � > 0 (for x > 0)
⇒ and E � ∝ De − Dh
⇒ n� �= p� (but still n� � p� )
⇒ and |n� − p� | ∝ De − Dh
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 10-11
hυ
n
x
0
p',n'
ε'
++
p'
ε'
n'
-
-
x
0
Je, Jh
Jh(diff)
Je(drift)
0
Jt=0 everywhere
x
Je(diff)
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 10-12
Solution
Step 1. Minority carrier flow problem (for x > 0):
d2 p�
p�
−
=0
dx2 L2h
with
Lh =
√
Dh τ
solution of the form:
p� = A exp
x
−x
+ B exp
Lh
Lh
B.C. at x = 0:
dp�
gl 1
= Jh(0) = −Dh |x=0
2
q
dx
Then:
p� =
g l Lh
−x
exp
2Dh
Lh
This works for x ≥ 0 because p� has to be continuous.
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 10-13
Step 2. Hole current:
Assuming Jh(drif t) � Jh(dif f )
dp� qgl
−x
Jh � −qDh
=
exp
dx
2
Lh
Step 3. Total current:
Jt = 0 everywhere
Step 4. Electron current:
Je = −Jh = −
qgl
−x
exp
2
Lh
Step 5. Electron profile:
n� � p� =
g l Lh
−x
exp
2Dh
Lh
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 10-14
Step 6. Electron diffusion current:
dn�
qgl De
−x
Je(dif f ) = qDe
=−
exp
dx
2 Dh
Lh
Step 7. Electron drift current:
Je (drif t) = Je − Je(dif f )
=
qgl De − Dh
−x
exp
2
Dh
Lh
Note: if De = Dh ⇒ Je(drif t) = 0
Step 8. Average velocity of hole diffusion:
vhdiff
Jhdiff (x)
Jh(x)
Dh
=
=
� �
qp(x)
qp (x) Lh
independent of x.
[will use when deriving I-V characteristics of pn junction diode]
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 10-15
Now verify assumptions
�
�
Step 9. Verify quasi-neutrality: | p −n
|�1
p�
Compute E � from Je (drif t):
E� =
Je (drif t) kT gl De − Dh
−x
=
exp
qµe no
q 2no De Dh
Lh
From Gauss’ law, get difference between n� and p� :
p� − n� = −
�kT gl De − Dh
−x
exp
q 2no 2Lh De Dh
Lh
Then
p� − n�
LD 2 De − Dh
|
|
=
(
)
p�
Lh
De
If characteristic length of problem is much longer than LD (Debye
length), quasi-neutrality applies in minority-carrier-type situations.
Put numbers: for ND = 1016 cm−3 , LD ∼ 0.04 µm, Lh ∼ 400 µm,
and (LD /Lh)2 ∼ 10−8.
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6.720J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 10-16
Step 10. Verify Jh(drif t) � Jh(dif f )
Jh(drif t)
qµhp� E �
1 p� De − Dh
|
|=
|=|
dp�
Jh(dif f )
2 no De
−qDh dx
as good as low-level injection
Step 11. Limit to injection to maintain LLI: p� (0) � no
gl �
2Dhno
Lh
Step 12. Verify linearity between v drift and E �
At x = 0 (worst point):
µe E � =
gl De − Dh
De − Dh
�
2no Dh
Lh
∼ 1000 cm/s � vsat.
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 10-17
� Example 2: Diffusion and surface recombination in
a ”short” or ”transparent” bar
Uniform doping: Eo = 0; static conditions:
∂
∂t
=0
Bar length: L � Lh; S = ∞ at bar ends.
hυ
S=
S=
n
-L/2
0
L/2
x
0
L/2
x
L/2
x
p'
-L/2
Je, Jh
Jh(diff)
Je(drift)
Jt=0 everywhere
-L/2
0
Je(diff)
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6.720J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 10-18
Length scales of minority-carrier situations
� Diffusion Length: mean distance that a carrier diffuses in a bulk
semiconductor before recombining
Ldiff =
√
Dτ
Ldiff strong function of doping:
Minority carrier diffusion length (cm)
1E+0
T=300 K
1E-1
Le
Lh
1E-2
1E-3
1E-4
1E-5
1E+14
1E+15
1E+16
1E+17
1E+18
1E+19
1E+20
1E+21
doping level (cm-3)
� Sample size, L
• If L � Ldiff , Ldiff is characteristic length of problem
• If L � Ldiff , L is characteristic length of problem
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6.720J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 10-19
Key conclusions
• Minority-carrier type situations dominated by behavior of mi­
nority carriers: diffusion, recombination and drift.
• Two characteristic lengths in minority-carrier type situations
dominated by diffusion and recombination:
√
– diffusion length, L = Dτ , average distance that a carrier
diffuses in a bulk semiconductor before recombining;
– sample size, L
– whichever one is smallest, L or Ldiff , dominates behavior of
minority carriers.
• Minority-carrier type situations called that way because:
– length and time scales of problem dominated by minority
carrier behavior (diffusion, recombination, and drift)
– role of majority carriers is to preserve quasi-neutrality and
total current continuity
• Order of magnitude of key parameters in Si at 300K:
– Diffusion length: Ldif f ∼ 0.1−1000 µm (depends on doping
level).
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 10-20
Self-study
• Work out example 2: diffusion and surface recombination in a
”short bar” (§5.6.2)
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].