Let us turn our attention to what happens to the electrons and
holesa, once they have been injected across a forward-biased
junction. We will concentrate just on the electrons which are
injected into the p-side of the junction, but keep in mind
that similar things are also happening to the holes which
enter the n-side.
As we saw a while back, when electrons are injected across a
junction, they move away from the junction region by a
diffusion process, while at the same time, some of them are
disappearing because they are minority carriers (electrons in
basically p-type material) and so there are lots of holes
around for them to recombine with. This is all shown
schematically in Figure 1.
It is actually fairly easy to quantify this, and come up
with an expression for the electron distribution within the
p-region. First we have to look a little bit at the
diffusion process however. Imagine that we have a series of
bins, each with a different number of electrons in them. In
a given time, we could imagine that all of the electrons
would flow out of their bins into the neighboring
ones. Since there is no reason to expect the electrons to
favor one side over the other, we will assume that exactly
half leave by each side. This is all shown in Figure 2. We will keep things simple and only look
at three bins. Imagine I have 4, 6 , and 8 electrons
respectively in each of the bins. After the required
"emptying time," we will have a net flux of exactly one
electron across each boundary as shown.
Now let's raise the number of electrons to 8, 12 and 16
respectively (the electrons may over lap some now in the
picture.) We find that the net flux across each boundary is
now 2 electrons per emptying time, rather than one. This
leads us to a rather obvious statement that the flux of
carriers is proportional to the gradient of their
density. This is stated formally in what is known as
Fick's First Law of Diffusion:
Flux=-
D
e
ddxnx
Flux
D
e
x
n
x
(1)
Where
De
De
is simply a proportionality
constant called the
diffusion
coefficient. Since we are talking about the motion of
electrons, this diffusion flux must give rise to a current
density
Je
diff Je
diff .
Since an electron has a charge
-q
q
associated with it,
J
e
diff
=q
D
e
ddxn
J
e
diff
q
D
e
x
n
(2)
Now we have to invoke something called the
continuity
equation. Imagine we have a volume
V V which is filled with some
charge
Q Q. It is fairly
obvious that if we add up all of the current density which
is flowing out of the volume that it must be equal to the
time rate of decrease of the charge within that volume. This
ideas is expressed in the formula below which uses a
closed-surface integral, along with the
all the other integrals to follow:
∮SJdS=-ddtQ
S
S
J
t
Q
(3)
We can write
QQ as
Q=∮VρvdV
Q
V
V
ρ
v
(4)
where we are doing a volume integral of the charge density
ρρ over the volume
VV. Now we can use Gauss'
theorem which says we can replace a surface integral of a
quantity with a volume integral of its divergence:
∮SJdS=∫VdivJdV
S
S
J
V
V
J
(5)
So, combining
Equation 3,
Equation 4
and
Equation 5, we have (note we are still
dealing with surface and volume integrals):
∫VdivJdV=-∮VddtρdV
V
V
J
V
V
t
ρ
(6)
Finally, we let the volume
VV
and we have the differential form of the continuity equation
divJ=-ddtρ
J
t
ρ
(7)
Now let's go back to the electrons in the diode. The
electrons which have been injected across the junction are
called excess minority carriers, because they
are electrons in a p-region (hence minority) but their
concentration is greater than what they would be if they
were just in a regular sample of p-type material. We will
designate them as
n'
n'
, and since they could change with
both time and position we shall write them as
n
'
xt
n
'
x
t
. Now there are two ways in which
n
'
xt
n
'
x
t
can change with time. One would be if we were to
stop injecting electrons in from the n-side of the
junction. A reasonable way to account for the decay which
would occur if we were not supplying electrons would be to
write:
ddt
n
'
xt=-
n
'
xt
τ
r
t
n
'
x
t
n
'
x
t
τ
r
(8)
Where
τr
τr
called the
minority carrier
recombination lifetime. It is pretty easy to show
that if we start out with an excess minority carrier
concentration
no
' no
' at
t=0
t
0
, then
n
'
xt
n
'
x
t
will goes as
n
'
xt=ⅇ-1
τ
r
n
'
x
t
-1
τ
r
(9)
But, the electron concentration can also change because of
electrons flowing into or out of the region
xx. The electron concentration
n
'
xt
n
'
x
t
is just
ρxtq
ρ
x
t
q
. Thus, due to electron flow we have:
ddt
n
'
xt=1qddtρxt=1qdivJxt
t
n
'
x
t
1
q
t
ρ
x
t
1
q
J
x
t
(10)
But, we can get an expression for
Jxt
J
x
t
from
Equation 2.
Reducing the divergence in the
above
equation to one dimension (we just have a
∂∂xJ
x
J
) we finally end up with
ddt
n
'
xt=
D
e
d2dt2
n
'
xt
t
n
'
x
t
D
e
t
2
n
'
x
t
(11)
Combining
Equation 11 and
Equation 8 (electrons will, after all, suffer
from both recombination and diffusion) and we end up with:
ddt
n
'
xt=
D
e
d2dt2
n
'
xt-
n
'
xt
τ
r
t
n
'
x
t
D
e
t
2
n
'
x
t
n
'
x
t
τ
r
(12)
This is a somewhat specialized form of an equation called the
ambipolar diffusion equation. It seems kind of
complicated but we can get some nice results from it. Let's
see what we can do with this.
For anything we will be interested in, we will only look
at steady state solutions. This means that
the time derivative on the LHS of equation 12 is zero, and
so we have (letting
n
'
xt
n
'
x
t
become simply
n
'
x
n
'
x
since we no longer have any time variation to
worry about)
d2dt2
n
'
x-1
D
e
τ
r
n
'
x=0
t
2
n
'
x
1
D
e
τ
r
n
'
x
0
(13)
Let's pick the not unreasonable boundary conditions that
n
'
0=
n
0
n
'
0
n
0
(the concentration of excess electrons just at
the start of the diffusion region) and
n
'
x→0
n
'
x
0
as
x→∞
x
(the excess carriers go to zero when we get far
from the junction) then
nx=
n
0
ⅇ-x
D
e
τ
r
n
x
n
0
x
D
e
τ
r
(14)
The expression in the radical
D
e
τ
r
D
e
τ
r
is called the
electron diffusion
length,
Le
Le
, and gives us some idea as to how
far away from the junction the excess electrons will exist
before they have more or less all recombined. This will be
important for us when we move on to bipolar transistors.
Just so you can get a feel for some numbers, a typical value
for the diffusion coefficient for electrons in silicon would be
D
e
=25cm2sec
D
e
25
cm
2
sec
and the minority carrier lifetime is usually around a
microsecond. Thus
L
e
=
D
e
τ
r
=25×10-6=5×10-3cm
L
e
D
e
τ
r
25
10
-6
5
10
-3
cm
(15)
which is not very far at all!
