The reason for calling the proportionality constant
Isat
Isat
will become obvious when we consider
reverse bias. Let us now make
Va
Va
negative instead
of positive. The applied electric field now
adds in the same direction to the built-in field. This means the
barrier will increase instead of decrease,
and so we have what is shown in Figure 1. Note
that we have marked the barrier height as
q
V
bi
-
V
a
q
V
bi
V
a
as before. It is just that now,
Va
Va
is negative, and so the barrier is
bigger.
Remember, the electrons fall off exponentially as we move up in
energy, so it does not take much of a shift of the bands before
there are essentially no electrons on the
n-side with enough energy to get over the barrier. This is
reflected in the diode equation where, if we let
Va
Va
be a negative number,
ⅇq
V
a
kT
q
V
a
k
T
very quickly goes to zero and we are left with
I=-
I
sat
I
I
sat
(1)
Thus, while in the forward bias direction, the current increases
exponentially with voltage, in the reverse direction it simply
saturates at
-
I
sat
I
sat
. A plot of
I I as a
function of voltage or an
I-V characteristic curve
might look something like
Figure 2.
In fact, for
real diodes (ones made from
silicon)
Isat
Isat is such a small value (on the order of
10-10
10
-10
amps) that you can not even see it on most common
measuring devices (oscilloscope, digital volt meter etc.) and if
you were to look on a device called a
curve tracer
(which you will learn more about in Electronic Circuits [ELEC
342]) what you would really see would be something like
Figure 3.
We see what looks like zero current in the reverse direction,
and in fact, what appears to be no current until we get a
certain amount of voltage across the diode, after which it very
quickly "turns on" with a very rapidly increasing forward
current. For silicon, this "turn on" voltage is about 0.6 to
0.7 volts.
Digital volt meters (DVM's) use this characteristic for their
"diode check" function. What they do is, when the "red" or
positive lead is connected to the p-side (anode, or arrow in the
diagram) and the "black" or negative lead is connected to the
n-side (cathode, or bar in the diagram) of a diode, the meter
attempts to pass (usually) 1 mA of current through the diode.
If the 1 mA of current is allowed to flow, the meter then
indicates the amount of forward voltage developed across the
diode. If it reads something like 0.673 volts, then you can be
pretty sure the diode is OK. Reverse the leads, and the diode
is reverse biased, and the meter should read "OL" (overload) or
something like that to indicate that no current is flowing.
The diode equation is usually approximated by two somewhat simpler
equations, depending upon whether the diode is forward or
reverse biased:
I≈0if
V
a
<0
I
sat
ⅇq
V
a
kTif
V
a
>0
I
0
V
a
0
I
sat
q
V
a
k
T
V
a
0
(2)
For reverse bias, as we said, the current is essentially nil.
In the forward bias case, the exponential term quickly gets much
larger than unity, and so we can forget the "-1" term in the
diode equation. Remember, we said that
kT
k
T
at room temperature had a value of about 1/40 of an
eV, so
qkT≈40V-1
q
k
T
40
V
-1
, this means we can also say for forward bias that
I=
I
sat
ⅇ40
V
a
I
I
sat
40
V
a
(3)
From this equation it is easy to see that only a small positive
value for
Va
Va
is needed in order to make the
exponential much greater than unity.
Now let's connect this "ideal diode equation" to the real world.
One thing you might ask yourself is "How could I check to see if
an actual diode follows the equation given here?" As we said,
Isat
Isat
is a very small current, and so trying
to do the reverse test is probably not going to be successful.
What is usually done is to measure the diode current (and
forward voltage) over several orders of magnitude of current.
While the current can vary by many orders of
magnitude, the voltage is more or less limited to values between
0 and 0.6 to 0.7 volts, not by any fundamental process, but
rather simply by the fact that too much forward current will
burn up the diode.
If we take the natural log of both sides of the second piece of
Equation 2, we find:
lnI=ln
I
sat
+q
V
a
kT
I
I
sat
q
V
a
k
T
(4)
Thus, a plot of
lnI
I
as a function of
Va
Va
should yield a straight line with a
slope of
qkT
q
k
T
, or 40.
Well, I went into the lab, grabbed a real diode and made some
measurements. Figure 4 is a plot of the natural
log of the current as a function of voltage from 0.05 to 0.70
volts. Included with this plot, is a linear curve fit to the
data which is plotted as a dotted line. The linear fit goes
through the data points quite nicely, so the current is surely
an exponential function of the applied voltage! From the
expression for the best fit, which is printed above the graph,
we see that
ln
I
sat
=-19.68
I
sat
-19.68
. That means that
I
sat
=ⅇ-19.68=2.89×10-9
I
sat
-19.68
2.89-9
amps, which is indeed a very small current. Look at
the slope however. Its supposed to be 40, and yet it turns out
to be slightly more than 20! This comes about because of
some complex details of exactly what happens to the electrons
and holes when they cross the junction. In what is called the
diffusion dominated situation electrons and holes
are injected across the junction, after which they diffuse away
from the junction, and also recombine, until eventually they are
all gone. This is shown schematically in Figure 5. The other regime is called
recombination dominated and here, the majority of
the current is made up of the electrons and holes recombining
directly with each other at the junction. This is shown in
Figure 6. For recombination dominated
diode behavior, it turns out that the current is given by
I=
I
sat
ⅇq
V
a
2kT
I
I
sat
q
V
a
2
k
T
(5)
In general, a particular diode might have a combination of these
two effects going on, and so people often use a more general
form for the diode equation:
I=
I
sat
ⅇq
V
a
nkT
I
I
sat
q
V
a
n
k
T
(6)
where
nn is called the
ideality factor and is a number somewhere between 1
and 2. For the diode which gave the data for our example
n=1.92
n
1.92
and so most of the current is dominated by
recombination of electrons and holes in the depletion region.
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