Lecture #9 OUTLINE • Continuity equations • Minority carrier diffusion equations • Minority carrier diffusion length • Quasi-Fermi levels Read: Sections 3.4, 3.5
Download ReportTranscript Lecture #9 OUTLINE • Continuity equations • Minority carrier diffusion equations • Minority carrier diffusion length • Quasi-Fermi levels Read: Sections 3.4, 3.5
Lecture #9
OUTLINE • Continuity equations • Minority carrier diffusion equations • Minority carrier diffusion length • Quasi-Fermi levels Read: Sections 3.4, 3.5
Derivation of Continuity Equation
• Consider carrier-flux into/out-of an infinitesimal volume: Area
A
, volume
Adx
J
N (
x
)
J
N (
x
+
dx
)
Adx
n
t
dx
1
q
J N
(
x
)
A
J N
(
x
dx
)
A
n
n Adx
Spring 2007 EE130 Lecture 9, Slide 2
Continuity Equations:
J N
(
x
n
t dx
) 1
q
J N
(
x
)
J N
x
(
x
)
J N
x
(
n x
)
n dx
n
t
p
t
1
q
1
q
J N
x
(
x
)
J P
x
(
x
)
n
n
p
p
G L
G L
Spring 2007 EE130 Lecture 9, Slide 3
Derivation of Minority Carrier Diffusion Equation
• •
The minority carrier diffusion equations are derived from the general continuity equations, and are applicable only for minority carriers.
Simplifying assumptions:
– The electric field is small, such that
J N
q
n n
qD N
n
x
qD N
n
x J P
q
p p
qD P
p
x
qD P
p
x
in p-type material in n-type material –
n
0 and
p
0 are independent of
x
(uniform doping) – low-level injection conditions prevail Spring 2007 EE130 Lecture 9, Slide 4
• Starting with the continuity equation for electrons:
n
t
n
0
t
1
q
J N
x
(
n
x
) 1
q
x
n
n
G L
qD N
n
0
x
n
n
n
G L
n
t
D N
2
n
x
2
n
n
G L
Spring 2007 EE130 Lecture 9, Slide 5
Carrier Concentration Notation
•
The subscript “n” or “p” is used to explicitly denote n-type or p-type material, e.g.
p
n is the hole (minority-carrier) concentration in n type material
n
p is the electron (minority-carrier) concentration in n-type material •
Thus the minority carrier diffusion equations are
t n p
D N
2
x
2
n p
n n p
G L
Spring 2007
p n
t
D P
2
p n
x
2 EE130 Lecture 9, Slide 6
p n
p
G L
Simplifications (Special Cases)
• • •
Steady state:
n p
t
No diffusion current:
D
0
p n
t N
2
n p
x
2 0 0
D P
2
p n
x
2 0
No R-G:
n p
n
0
p n
p
0 •
No light:
G L
0 Spring 2007 EE130 Lecture 9, Slide 7
Example
•
Consider the special case:
– constant minority-carrier (hole) injection at
x
=0 – steady state; no light absorption for
x
>0
p n
( 0 )
p n
0 0
D P
2
p n
x
2
p n
p
2
p n
x
2
p n D
P
p
p n L
P 2
L
P is the
hole diffusion length
:
L P
D P
p
Spring 2007 EE130 Lecture 9, Slide 8
The general solution to the equation is
p n
(
x
)
Ae
x
/
L P
Be x
/
L P
2
p n
x
2
p n L
P 2 where
A, B
are constants determined by boundary conditions:
p n
( ) 0
B
0
p n
( 0 )
p n
0
A
p n
0 Therefore, the solution is
p n
(
x
)
p n
0
Ae
x
/
L P
Spring 2007 EE130 Lecture 9, Slide 9
Minority Carrier Diffusion Length
• Physically,
L
P and
L
N represent the average distance that minority carriers can diffuse into a sea of majority carriers before being annihilated.
• Example:
N D
=10 16 cm -3 ; p = 10 -6 s Spring 2007 EE130 Lecture 9, Slide 10
Quasi-Fermi Levels
• Whenever
n =
p
0
, np
n i 2
. However, we would like to preserve and use the relations:
n
n i e
(
E F
E i
) /
kT p
n i e
(
E i
E F
) /
kT
• These equations imply
np = n i 2
, however. The solution is to introduce two
quasi-Fermi levels F
N and
F
P such that
n F N
n i e
(
F N E i
E i
) /
kT kT
ln
n n i
p F P
n i e
(
E i
F P
) /
kT E i
kT
ln
p n i
Spring 2007 EE130 Lecture 9, Slide 11
Example: Quasi-Fermi Levels
Consider a Si sample with
N
D = 10 17 cm -3
n
=
p
= 10 14 cm -3 .
and
What are p and n ?
What is the np product ?
Spring 2007 EE130 Lecture 9, Slide 12
•
Find F
N
F N
E i
and F
P
kT
ln
:
n n i
F P
E i
kT
ln
p n i
Spring 2007 EE130 Lecture 9, Slide 13
Summary
• The
continuity equations
are established based on conservation of carriers, and therefore are general:
n
t
1
q
J N
x
(
x
)
n
n
G L
p
t
1
q
J P
x
(
x
)
p
p
G L
• The
minority carrier diffusion equations
are derived from the continuity equations, specifically for minority carriers under certain conditions (small
E
field, low-level injection, uniform doping profile):
n p
t
D N
2
n p
x
2
n p
n
G L
p n
t
D P
2
p n
x
2
p n
p
G L
Spring 2007 EE130 Lecture 9, Slide 14
• The
minority carrier diffusion length
is the average distance that a minority carrier diffuses before it recombines with a majority carrier:
L N
D N
n L P
D P
p
• The
quasi-Fermi levels
can be used to describe the carrier concentrations under non-equilibrium conditions:
F N
E i
kT
ln
n n i
F P
E i
kT
ln
p n i
Spring 2007 EE130 Lecture 9, Slide 15