So I feel very excited to share with you one of my favorite papers
and us this paper, which is titled
Detailed Balance Limit
of Efficiency of p-n Junction based Solar Cells
and uh... this is not just my favorite paper
it's in fact one of the
uh... most highly cited papesr in the field of semiconductors this paper has
been cited uh more than uh two thousand times so that speaks to the quality of
the paper
and uh... sometimes this paper is also known as the Shockley-Queisser
limit
because of uh... the authors of these papers were uh... william Shockley
and uh Hans Queisser
who used to work for William Schockley at Shockley Transistor at that
time
And let me just read through the abstract of this paper to give you some
context
So this paper says that uh... these uh... these two guys William Shockley
and Hans Queisser,
they are interested in finding the upper theoretical limit of efficiency conversion
of a p-n junction based solar cell.
And they apply this principle of detailed balance to
find this limit and this principle of detailed balance is
is uh something which is uh
used in the field of thermodynamics and so they're applying essentially
a principle,
a thermodynamic principle, to find the theoretical limit
of efficiency conversion for a p-n junction based
solar cells.
And they say that this is uh for an ideal case in which the only recombination
mechanism
is uh... of between the electron and holes
is that of radiative. So the only recombination mechanism that they're
taking into account is uh radiative recombination.
And uh... then further they're going to say that uh... the sun and the cell
are assumed to be black bodies
with temperature of 6000 Kelvin and 300 Kelvin so they have a
assumed uh
the sun and the cell uh... to be the radiation coming from the sun and the cell
to be uh... equal to that of a black body with these temperatures.
So if they do that, then the maximum efficiency that they find
is uh... of thirty percent
and that corresponds to energy bandgap of uh 1.1 (eV)
which is very close to what you
will get uh... in uh... silicon.
So to give you some, to give you some context, this uh... paper it was
uh... published in 1961
and uh... efficiencies of solar cells
the first uh
um... you know and uh... good solar cell which was demonstrated by Bell Labs was
demonstrated in 1954 and had an efficiency of around 6%.
And uh... more and more cells uh
were uh... which are published or which were being reported around
that period still had an efficiency of 9-10%.
And what this paper, this audacious paper from Shockley and Hans Quiesser
they were saying that these cells can go up to thirty percent efficiency
so they had, you know, they got some slack for giving this prediction, and they had to
essentially explain that
uh...the actual junctions they do not match these efficiencies, they had to
give some explanation that why is this uh... why this uh discrepancy
between uh
uh between the obversed efficiency and what they were predicting.
So
I'm showing here what the other people at that time were doing. So
these were, this was the approach that the peers of Shockley and Queisser were taking
to determine again
these maximum efficiencies.
And the reason why there are a lot of interest in determining these maximum
efficiencies was that
these solar cells were being considered for application into space,
in fact Russia had just launched a satellite which used these solar cells
uh... as the power source
so there was a large amount of interest in finding out what would be
the maximum efficiency that these asolar cells can achieve.
So most of the peers, most of the colleagues of Shockley and Queisser,
they were approaching this problem in this way: so they were essentially
considering these p-n junction based solar cells made up of certain
materials, let's say silicon in this case,
and they were considering all the loss mechanisms that
can occur in these cells so ideally you want these electrons
and holes to be collected at these uh
uh... and these two contacts, but there might be a stray
carrier which gets collected at that wrong contact, so there might be a
loss mechanism which is occurring at the contact.
Similarly, instead of being collected, these electron and hole pairs
which are generated, instead of being collected at the contacts
they can recombine with each other and they can emit out a photon.
So most people you know they were using material properties to calculate
uh... this uh
uh... this uh... recombination mechanism, but this thing where gets
complicated that this photon that is emitted can essentially get reabsorbed
back in the cell
and it can generate another hole and electron pair
and further this can emit back a photon which can get reabsorbed
as well.
So this is, you know, if you approach this problem from a materials
perspective, you immediately see the fallacies of that so
what Shockley and Queisser did, they said, you know, let's not
let's not take this approach and they essentially said, you know, we will
not approach uh... this uh problem in this way,
and in fact approached this uh... problem using this uh method of detailed
balance which uh... is uh just to approach this problem from a thermodynamic
perpective.
But they did make certain assumptions to uh... certain assumptions
to solve this uh... problem.
So the first assumption that
they made was they assumed that uh... the concentration of sunlight was
uh... one sun
so they did not assume that you are
uh... using any concentrated optics and uh...
uh... they assumed that the concentration was just equal to one sun. Another
assumption that they made that they uh ... which I uh mentioned in the abstract that they
assumed that the the sun and the cell, they act as ideal blackbodies,
the sun at a temperature of 6000 Kelvin
and the cell at a temperature of 300 Kelvin.
Another assumption they made was that absorption in this cell, it essentially
had a step profile of a step function
and what I mean by that is that all the photons which have
energy
less than the bandgap, they don't get absorbed in the cell at all,
and all the photons which have uh... energy greater than or equal to the
bandgap,
they have an absorption coefficient of one, or you know, all of these photons which have
energy greater than the bandgap, they get completely absorbed, and this absopriton
function is like a step function.
Another assumption they made was this uh
carriers in this semiconductor, they have infinite mobility, so if these
electron and hole pairs are generated, they
they will essentially, they have the capability to diffuse, uh
until these contacts, where they will be collected.
And uh... another assumption which I stated in the abstract is that the only
mechanism that they assume for recombination was this radiative
recombination.
And there might be other mechanism which can happen, for exmple if you have
traps in you semiconductor
there's a Shockley-Read-Hall mechanism which uh... can also give you recombination,
But they assumed that you have an ideal semiconductor
and the only
recombination mechanism that you can have is radiative recombination.
So before we get into, your know, before we get into the actual
Shockley-Queisser limit, the paper actually presented more
like a fun exercise, so they said:
What would be the limit of the solar cell if the temperature of
the cell was essentially 0 Kelvin? So before actually evaluating the
problem for a realistic case where the temperature of the is uh
300 Kelvin, which is closer to room temperature,
they first said "Let's evaluate this problem for the case of
temperature of the cell being 0 Kelvin."
And they, uh, in this figure that I'm borrowing from the paper, it uh
essentially also assumes this funny-looking geometry of the cell.
So they assume the cell to be circular in shape, where I have this
uh... p region on the outside and this n region on the inside.
and uh... they assume the solar radiation to be coming from
this disk, so they assume the solar radiation to be coming
from this disk which is at a temperature equal to the temperature of the sun.
And this is a very idealized case, so they want to maximize the efficiency
so they're assuming that the
radiation is the incident on all the surfaces of the cell; usually it's only
incident on uh, on your
on one particular side of the cell which is facing the sun.
So this is a.. this is uh... idealized case. So they want to calculate the maximum
efficiency uh that you can get in this case. So what would be the efficiency
in this case? The efficiency would be the total
uh generated photon energy divided by the input power.
So the way you can calculate the total generated photon energy would be to calculate
the number of photons absorbed, and that's given by
that's denoted by this symbol Q_S,
and the way Shockley and Queisser calculated that is that they assumed
this step function far uh, for the absorption, so they assumed that all
the photons which have a frequency which corresponds to an energy greater than the
bandgap
they'll get completely absorbed, and all the photons which have energy which is
uh, less than the bandgap, so this is
energy and all the photons which have energy which is less than the bandgap,
they will not get absorbed. So this is an ideal step function
profile for the absorption function
and then all those photons which are absorbed, they are collected at the
energy which is equal to, equal to the bandgap of the semiconductor.
And this is divided by the input power contained in the spectrum. So how are these uh...
these Q_S and P_S calculated? So
they are calculated using our friend uh... Planck's Law, which we've derived in one of the
the earlier videos,
and this gives me the spectral irradiance, or the power which is contained
in the spectrum as a function of, as a function of the wavelength or the
frequency, and uh... as a function of the temperature off my black body, which ocurrs
over here.
So now the way I can calculate the number of photons absorbed is to
essentially
uh... take an integral of uh ... the number of incoming photons at each
frequency and integrate it over the frequency which corresponds to
the bandgap of the cell all the way to infinity. So if I perform this integral,
this gives me the total number of photons that would be absorbed.
And how do I calculate the input power? So the way I calculate the input power is
I integrate this power spectral density
over, uh, over the different wavelengths, so I perform this integral now going all the way
from the wavelength of zero, or frequency of zero, to
a frequency of um... a frequency of infinity.
And this is something that we have discussed previously that this is also
the... approximately the Stefan-Boltzmann Law.
And uh...
once you have these two values, you can essentially plug them in to this
equation and calculate the maximum efficiency.
And it's not a very hard calculation to do - you can buy a few lines of uh...
Matlab code to do so and uh... if you uh... if you uh... are so inclined and uh...
do uh... write up this Matlab code,
so this is what you get. So this is the the uh... ultimate efficiency which is uh
derived assuming that the cell is at, uh
is at zero Kelvin, and the sunlight is incident on the cell
from all directions
and uh...
and the efficiency you get this case, the maximum efficiency you get, is
approximately
uh... forty four percent
and uh... it occurs
when you have a bandgap of your absorbing material, a bandgap of
you semiconductor,
it's approximately 1.08 eV.
So this efficiency is much higher than what is the Shockley-Queisser limit.
The Shockley-Queisser limit is in fact around uh
30%, as it was stated in the
beginning of the paper,
and this additional loss comes from,
uh... the fact that the temperature of the
cell is not 0 Kelvin, in fact if you assume it to be
300 Kelvin, then this cell would act as a black body
as well and it will emit out this radiation of its own.
And we need to take into that, we need to take that into account,
uh... to derive this, uh, Shockley-Queisser limit. And I'm out of time in this
video, but I'll complete the derivation in the next video. So see you soon.