Shown here is the JV relationship for an ideal diode.
And this is also plotted in this chart over here, so
this axis plots the current density as a function of the voltage.
So what you see when
there's no applied voltage, there's VA equal to 0.
then this first term in this equation goes to 0.
And you get a current which is equal to the short circuit current.
Similarly, you see that the cell here has been optimized for solar cell operation.
So you see that when when VA is large, this
first term is large, so this Jsc remains a constant.
And then after
a while this J, this first term starts to take, increase exponentially.
And that results in this exponential increase in the short circuit current.
And then you reach a point where this first
term cancels out the current due to this generation.
And we reach a point which is called the open circuit open circuit voltage,
at which the total current is 0.
So this Jsc and Voc are important parameters
to keep in mind for any solar cell.
Another important parameter to keep in mind is
the maximum power.
So, we could operate this solar cell at any of these voltages.
The amount of power which we would extract, extract at each of
these points would be essentially represented
by this area under the strokes.
So for example if i'm operating on this
point number one i'm operating at this much current,
and i'm operating at this much voltage.
And the product of two, which represents the power that can,
I can extract from the solar cell.
Similarly, if i'm operating at this other point, this
area under the green triangle represents the total power output.
So i want to maximize this area under this
.
And that will occur at, the voltage at
that, that, condition occurs is typically the maximum
the maximum power point voltage, and the
correspondingly current is called the maximum power current.
Another important, parameter to keep in mind is the fill factor.
So, this fill factor is essentially defined as
this area at which the maximum part occurs, so Vmp into
Imp, and divided by the open circuit voltage and the short circuit current.
So this is a also important parameter
which is sometimes used to characterize solar cells.
And if it, if you have either cell, no, not many sources of recombination or
not many non-idealities, then this fill factor is typically
very high, close to 90% or or even higher.
So this in this video what I want to study is that what happens if you
have traps present inside this depletion region.
So how does the IV characteristics change as a
result of that, and what happens to these parameters?
So, shown here is a different recombination mechanism that can go on.
So, this, I realized, the relationship that i derived for the diode, that
already took into account this radiative recombination in the quasi-neutral region.
But there could be even other other recombination mechanisms,
such as this recombination with the help of a trap.
So for example, you have the trap level present
inside the band gap, and your electron and hole pair could
recombine with each other with the help of this trap level.
And the statistics of that is given by this Shockley-Read-Hall formula.
And this is something that we will concentrate on this video.
There could be other re-combination atoms such as Auger where this electron and hole pair
recombine, and we give that extra energy either to this another electron or
this another hole,
which subsequently thermalize and you know,
give that energy back to the lattice.
So, you know the different different recombination uh,
mechanisms. For this video I want to
focus on this
Shockley-Read-Hall
mechanism and then maybe we can generalize it
to other forms of recombination and other non-idealities as well.
So this Shockley-Read-Hall recombination is given by
this formula, where it's proportional to the
number of traps, to the capture cross-section
of the traps, to the thermal velocity.
in terms of carrier dependence, it depends upon the product of np over here.
And then in denominator it also has this concentrations of electron and hole in
these terms over here.
And it also depends upon the position of this trap energy level.
So this recombination is, is dependent upon where this
trap energy level is located within this bandgap.
Usually our recombination is maximum when this trap energy level
Is located right at the center of the bandgap,
or located close to the intrinsic energy level.
So I want to take
a pessimistic case over here and want to maximize this Shockley-Read-Hall rate
and that would give me the maximum amount of recombination that could occur.
So usually I have this p-n junction and i'm generating,
generating these electron and hole pairs in my depletion region.
And they are separated due to the electric field present
in this
collected out in the form of useful electricity.
But if I have these trap levels which are preset
in the, within the band gap of the semiconductor, then instead of, instead of
getting extracted out, these electrons are recombined, with each other in the depletion region.
And that would result in a decrease of
the amount of useful useful electricity I collect.
So I want to, you know, I want to calculate the maximum
rate of this Shockley-Read, maximum rate at where this Shockley-Read-Hall
recombination can occur. So this rate is maximum when this trap
energy level lies close to the intrinsic energy level or this Et is equal to Ei.
So this this term over here bit, becomes equal to,
, it becomes equal to 1 in that case.
It's also maximum if you have equal number of electrons and holes present.
So the number of electrons and hole present are equal,
so they have a better chance of recombining in this depletion region.
So it is also maximum when this n and p are equal.
Also, I know that the product of my number of electrons and holes, I can express
this as a function of applied voltage by this relationship
where it varies exponentially as a function of applied voltage.
So in the case when this number of electrons and holes are equal, both of them
would be equal to, both of them would be equal to ni times exponent of qvA by 2kT.
So i'm just taking a square root of this formula.
So if
I substitute all that in this equation, so I'll substitute for np as ni squared
exponential function of qA, vA by kT, then i substitute by n and p by this formula.
So I get a relationship which describes the
recombination rate, as as given by this relationship.
Where I can express this rate of recombination by this formula,
where in the numerator I get qvA by kT
minus 1, and in the denominator I get qvA by 2kT minus
1. And this tau over here is is term that
bunches together the thermal velocity, the number of traps, and
their capture cross section.
So when this VA is large, this exponential term is
going to be much larger as compared to 1.
So I can,
, i can essentially simplify this further. And that gives me this
maximum rate of Shockley-Read-Hall recombination to be equal to,
to be equal to ni by 2tau
exponential of qVA by 2kT.
So now my current due to this Shock, due
to this Shockley-Read-Hall recombination, would be equal to this Shockley-Read-Hall
rate multiplied by the total multiplied by this depletion
width in which this in which this recombination can occur.
So this, current due to this Shockley-Read-Hall , recombination,
I can express that, by this relationship, where it
depends exponentially, so that J02 will will bunch around these terms,
and it depends exponentially on the applied
voltage, but this exponential dependence is given
by qVA by 2kT instead of qVA by kT for the ideal diode relationship.
So overall I can write down my total current.
I can write down my total current.
I can combine, I'll combine basically my my ideal diode equation.
So this would
be, this would be, have an exponential dependence of qVA by kT,
and then i'll have this term, due to
this due to the, due to the Shockley-Read-Hall recombination,
and that has exponential dependence of qVA by 2kT,
. And then i have my Jsc current over here.
So what does this, addition of this, addition of this
of this current do to my, to my diode IV characteristics?
So first thing I do or notice is that now I have of these two things
which are essentially you know, these are acting together to compensate for my Jsc.
So, what I see over here is that,
for a given applied voltage, I will get, due to these first two
terms, I will get a larger positive thing which is trying to cancel my Jsc.
So, what I immediately notice is that
my Voc in this case would essentially decrease.
If I consider the case when I have V applied equal to 0 then I don't
see much of change because of at V applied equal to 0 this term will become 0,
and this term will become, zero.
So this, this, this, overall current would still be Jsc.
So these are two my, these are two of my endpoints. Then in terms of this
exponential rise, usually I've only had this exponential term of qVA by kT.
But now I have term which is a half in terms
of that exponential part, so it's a term of qVA by 2kT.
So what that means is that this slope of this
I-V relationship wouldn't be as sharp, so I can expect,
you know, that I'll have something which will have a
lower slope, so instead of having this particular slope and this
exponential rise, i'll see a slightly
lower slope when this exponential rise happens.
So that will result in, you know, a I-V relationship which, which
might look something like this.
where I start off with Jsc but then I
have this this particular rise, which is not as sharp.
So, in terms of my efficiency, you can see,
clearly see that my Voc is decreasing, and also
since this is, this curve is not as sharp
when I now operate this operate at the maximum power condition,
I'll see that
my fill factor has now reduced, because my, I'll only
be able to occupy this area under this shaded rectangle.
versus my Voc product with Jsc would be higher,
so my fill factor has reduced.
My Voc has reduced, my Jsc doesn't change much.
So whenever I have these traps present in my depletion region, they lead to
primarily a degradation of Voc, and a degradation of the fill factor.