So I want to touch upon some basics of p-n junction or a p-n diode in this video.
So one, one way to think about this p-n junction
is to think about these individually doped p and n regions.
So I have these two semiconductor blocks.
One is a n-type and it has essentially donor atoms
which are added to this to the lattice
of the semiconductor
and they have this extra electron which is given to
the lattice, and they, and what it results in is
that, it results in this excess population of electrons, which
are present in the conduction band of this n-type semiconductor.
Similarly in p-type semiconductor, we're using the same analogy.
What we do over here is we add these acceptor atoms, which have one less
electrons as what is needed to complete the bonding in the lattice.
So that produces a absence of electrons, which are
represented as holes present in the valence band.
So now, a good way to think about how this p-n junction works
is to think about, when we have these individual p and n-type semiconductor,
essentially what I'm doing is I'm bringing
them, you know I'm bringing them closer and closer
and I'm bringing them in contact with each other.
So now what happens if I essentially, essentially put these, two in contact? So
let me think of in terms of these carriers.
So I have these electrons or excess electrons over here
so there are plenty of electrons in the n-type semiconductor.
While there're no electrons or you know, there are an absence of electrons
in the p-type semiconductor.
So these guys which are present in excess over here, they
would very much like to go into this uh p-type semiconductor.
Because, you know, if you think
about the excess of these electrons over here
and there'll be, they think they'll be more valued if they go to
some place where there's a need of
electrons or there is a absence of electrons.
So these are if you think in terms of electrons, these electrons want to jump
from this n-type to the p-type semiconductor.
Similarly, if I think of it in term of these holes,
so there are a lot of
these holes present in p-type semiconductor, while there
are no holes or, you know, way less
holes which are present in the n-type semiconductor.
So these holes, well they want to jump over to
the n-type semiconductor, where they think they'll be valued more.
So, this results in your immediate exchange
of carriers, so these electrons go from here to here.
And these holes go from p-type to the n-type semiconductor.
And finally we reach a state of equilibrium.
So let me explain why this equilibrium happens.
So when these electrons leave from these, from the n-type semiconductor and they
go to the p-type semiconductor, what they leave behind is these positively
charged donor atoms.
And remember that we had these donor atoms over here.
And acceptor atoms in the p-type semiconductor.
So what they leave behind is
this positively charged, positively charged donor atoms.
And similarly when these holes which jump from, from the p-type semiconductor
to the n-type semiconductor, what they leave behind is this acceptors
over here, which are negatively charged.
So they leave behind these negatively charged acceptor atoms.
And this happens in a certain region, which
is what is called as a depletion width.
So let me denote
this depletion rate on the p-type semi-connector with
this letter x with a subscript p.
Similarly, I can denote the depletion region in the n-type semiconductor with this
letter x and a subscript n. So now you see what has happened over
here, so I have these holes which are present in the p-type semiconductor.
They want to defuse to the n-type region.,
but now there's a security guard, this this
positively charged donor atom which are standing over here.
And these holes are positively charged as well,
so there's a repulsing, there's a repulsion between these
holes and these positively charged donor atoms.
And they keep you know, these these holes present in the p-type semiconductor.
They keep them in check and prevent them
to, you know, further diffusing into the n-type semiconductor.
Similarly, if I think in terms of these holes which are now present
in the n-type semiconductor, so they again want to go to the p-type semiconductor.
But, again, there's this security guard,
this negatively charged acceptor atoms,
and since these electrons are negatively charged as well,
so there's a repulsive force between these two
that prevents further diffusion of these carriers.
And this is what is called as we have reached, equilibrium state.
So now let me, let me think what happens in the in the equilibrium state.
What is the electric field?
What is the potential profile?
So if I think, let me start with first this in terms of this charge profile.
So if I think of this charge profile,
of the charge which is present in the semiconductor,
what I have just stated is that the charge in this this depletion region on the,
on the, p side, on the p-type is
essentially equal to this negative times the concentration
of the acceptors. And similarly the charge present on the
on the n-type is equal to this
positively charged donor atoms, which are left behind.
And, charge neutrality must hold true.
So basically, you know, what is our total charge present in the in this
region must be equal to the total charge which is present in the
in in this upper rectangle. So that gives me my first equation,
that is my charge neutrality,
which is this concentration of donors into the depletion width on the n-type region.
And this must be equal to the concentration of acceptors multiplied by,
multiplied by the depletion width in the p-type region.
So what I'm doing is essentially equating the
area under these two rectangles.
And that must hold true because overall there's no net
charge which was added or removed from this p–n junction.
And I've made this up futher approximation
that, you know, whatever is the charges present in just this depletion region.
There's no charge present outside the depletion region.
And this approximation is was is called as a depletion approximation.
So, so making this depletion approximation,
I can write the charge neutrality.
I can express that in terms of this equation.
So, now the next thing for
me to do is to relate this charge density or this charge profile.
And I can use some you know some basic physics I learned
in high school to relate it to the electric field and relate
it to the potential profile which is present in this p-n diode.
So how can I do that?
So I know from my high school physics that my
charge density is related to the electric field using
this relationship, where it's dependent on my electric field,
the derivative of my electric field is dependent upon the charge density
divided by the dielectric constant of my
semiconductor and the permittivity of the free space.
I also know that my electric field is related to my potential
profile by again this relationship. So what I can do is
integrate my charge profile and that will give me the profile of my electric field.
And then I can integrate my electric field
and that will give me the potential profile.
So when I integrate over this charge profile, so I have
this constant, constant charge density in my depletion region which is negative
in the p-type region and positive in the n-type region.
So when I integrate this rectangle, what I will get is this triangle over here.
And I'll get a maximum electric field,
which would be essentially equal to this total
charge present in the in the p-type region,
which would be equal to ND times XP.
And then using this
Poisson relationship, I'll get that maximum electric field to be this.
And which is also I can express that using this charge neutrality condition, in
terms of the total charge present in the n-type region and both are the same.
So, this is my this is my this my maximum electric field.
Let me correct this mistake over here. So that
should be Xn.
And then when I integrate so I see that I have a maximum maximum electric field
which just occurs at the junction, right
at the junction between the p and n-type semiconductor.
And this electric field, again it reduces as I further
travel into the n-type semiconductor. And you see that outside the depletion
region, outside the depletion region, there's no electric field.
And that's, we can also relate to that
by this charge profile, so these charged, total charge
which is present in the p and n
type region are equal and opposite to each other.
So if I think of the total charge present outside this depletion region,
that's again zero, so that makes sense that there would be no electric field
present outside this depletion region.
And now I can integrate this triangle further, so I can integrate this triangle.
So this triangle is essentially a linear relationship and when I get when I
integrate it I'll get a square relationship in terms of for my distance.
So when I integrate this electric field, what I'll get is this total
potential drop across my p-n junction. And I can denote that by this
symbol Vbi where it represents the built in voltage across my p-n junction.
And this Vbi is essentially given by this relationship where I get square in terms
of of my length, of my depletion width. And
I get a factor of 2 which comes from integrating this.
And this is essentially gives me this Vbi in terms of the, in terms of the doping
concentration and the depletion width. So now I have two relationships.
I have this is my equation number one. And this is my equation number two.
And I have
two variables, x n and x p.
And I have two equations, two variables, so
let me complete their solution in the next video.