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The continuity equation for particles conservation in semiconductors, focusing on electrons and holes. It provides the general equations for one-dimensional cases and simplifications for minority carrier diffusion with assumptions such as thermal r-g process and low-level injection. The document also includes examples of how excess carriers decay after turning off the light and determining the excess carrier concentration in a uniformly doped si.
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The continuity equation satisfies the condition that particlesshould be conserved! Electrons and holes cannot mysteriouslyappear or disappear at a given point, but must be transported to orcreated at the given point via some type of carrier action.Inside a given volume of a semiconductor,There is a corresponding equation for electrons.
etc.
otherslight
G
R thermal
diffusion
drift
t p
t p
t p
t p
t p
โ
x
x
x
p
x
p
x
x
q
(Flux
of holes)
Volume =
x
Area
โโ โ โ
โโ โ โ
โ
โค โฅ โฆ
โก โข โฃ
โ โ โ + โ =
โโ โ โโ
โโ โ โโ
โ โ โ + โ + โ = โ โ
โ
โ
โ โ
โ
etc.
light
G
R
thermal
p
p
p
etc.
light
G
R
thermal
p
p
)
(
)
(
t p
x
A
x
x
x
J
x
J
A q
x
J
q A
t p x A x x J
A q
x
J
q A
x
A
t p
Electric field
= 0 at the region of analysis
Equilibrium minority carrier concentrations are not functionsof position, i.e.,
n
0
n
0
x
p
0
p
0
x
Low-level injection
The dominant R-G mechanism is thermal R-G process
The only external generation process is photo generation
Apply the continuity equations to minority carriers, with thefollowing assumptions:
Consider electrons (for p-type) and make the followingsimplifications:
t
n
n
n
t
t n
t n
n
t n
x
n
n
n
x
n x
n x
qD
x n
qD
n
q
G
R
ฯ โ
ฮผ
โ
0
L
etc.
light
n
thermal
0
n
n
n
n
and
Consider an n-type Si uniformly illuminated such that theexcess carrier generation rate is
L
e-h pairs / (s cm
3
). Use
MCDE to predict how excess carriers decay after the light isturned-off. t
uniform
d/d
x
is zero; steady state
d/d
t
So, applying to holes,
p
t
L
ฯ
P
t
L
= 0; uniform
d/d
x
ฯ โ โ = โ ฯ โ โ = โ
t
p
p
p
t p
exp
so,
n
n
p
n
n
p
L
p
P
L
since
exp
ฯ
ฯ โ ฯ = > โ
G p t G t p
Consider a uniformly doped Si with
D
15
cm
โ
3
is illuminated
such that
p
n
10
cm
โ
3
at
x
= 0. No light penetrates inside Si.
Determine
p
n
x
). (see page 129 in text)
p p p p 0 n n
where
exp
ฯ
x
p
x
p
Solution is:
