Video Transcript
In this video, our topic is doped
semiconductors. We’re going to learn what it means
to dope a pure semiconductor sample, why we would do this, and we’ll also learn how
to describe the two main classes of doped semiconductors. Before we talk about this process
of doping though, let’s remind ourselves of what a semiconductor is and what a pure,
that is, undoped, sample looks like. Semiconductors, we can recall, sit
in between electrical insulators and conductors. Silicon is the most common material
used to make semiconductors. And if we look at this element on
an atomic scale, we know that every neutral atom of silicon has four valence
electrons or four electrons in its outermost shell.
This means that when many silicon
atoms are joined together in an orderly pattern called an atomic lattice, due to
electron sharing that takes place because of covalent bonds forming between adjacent
silicon atoms, atoms that are interior to the lattice, this one here is our example,
have a full set of electrons, eight of them, in their outermost shell. When an atom has a full complement
of valence electrons like this, it is unlikely to accept any more electrons from its
environment or give any electrons to that environment. In other words, it’s electrically
stable.
Now, in any realistically sized
atomic lattice, there will be millions, if not billions or trillions, of such
silicon atoms. And in that case, the interior
atoms, the ones surrounded by silicon atoms on all sides, would form the great
majority of the atoms in this lattice and therefore would determine its properties
overall. And specifically, we’re thinking of
its electrical properties, its ability to transmit charge. Silicon atoms that stably possess a
full complement of valence electrons provide no opportunity for electric charge to
make its way across the lattice. Set up this way then, our
semiconductor is behaving like an insulator. It’s not allowing electrical charge
to move through it.
If we want to increase the
conductive ability of our lattice, there are two ways we can do that. One way is to increase our
semiconductor’s temperature. By adding thermal energy to the
atoms of this lattice, we provide energy that’s needed for some of the electrons to
break the covalent bonds they’re in and become what are called free electrons, that
is, electrons not currently bound to any atom. Whenever a free electron is
created, it leaves behind a vacancy or an electron hole. The interaction between free
electrons and holes throughout the entire lattice raises the conductivity of our
semiconductor.
Say, for example, that once this
electron hole has opened up, a free electron from elsewhere in the lattice moves in
to fill that spot. And the electron that was liberated
to open up that vacancy in the first place may find a hole to occupy in the lattice
somewhere over here. In a pure semiconductor material
like we have here, which is all silicon, the number of free electrons is always
equal to the number of vacancies or electron holes. The interaction on these free
electrons and holes is what gives rise to the conductivity of our sample. And if we want to increase that
conductivity, say, by making more free electrons and more holes, then we can raise
the temperature of our semiconductor.
That, as we said, is one way we can
make our semiconductor more like a conductor. And the second way, which is the
main topic of this lesson, is by doing what’s called doping the semiconductor. When we do this, we add what are
called impurities to our pure semiconductor sample. Remember, we said earlier that
every silicon atom in this lattice has four valence electrons to it. That’s true when we’re working with
electrically neutral silicon.
This fact of having four valence
electrons had a significant effect on the behavior of our lattice when we fit all of
these atoms together. Specifically, we saw that, thanks
to electron sharing due to the formation of covalent bonds, an atom that started
with four valence electrons and that was interior to the lattice, like this atom of
silicon here, ended up with a full set, eight valence electrons, thanks to this
sharing.
But what if, in place of this
interior silicon atom, we put a different kind of atom, one that didn’t start out
with four valence electrons? Well, that’s what it means to add
an impurity to our semiconductor. Generally speaking, impurities that
we would use to dope a semiconductor would either have one more valence electron
than silicon — that is, they would have five — or one fewer, three. There are a number of instances of
atoms like this, but we’ll just look at one for each type. As an example of an atom with five
valence electrons, we can consider phosphorus and as an instance of one with three,
boron.
Now, for our discussion, it’s not
so important specifically what kind of element each of these atoms is, but rather
we’re most interested in the number of valence electrons each one has. It’s that number which will
determine the behavior of a semiconductor they’re a part of. Along these lines, there are
actually specific names for atoms that have four valence electrons or five or
three. An atom with four valence
electrons, like we saw silicon has, is called tetravalent, whereas one that has
five, like our phosphorus, is pentavalent. And boron is an example of a
trivalent atom, one that has three valence electrons. So, given that our pure
semiconductor sample is comprised entirely of tetravalent atoms, silicon in this
case, we’ll use penta- and trivalent atoms as our impurities.
To see how this works, we imagine
making a substitution. Imagine that we take an atom of
phosphorus and we replace the central silicon atom in our lattice with it. So basically, the silicon comes out
and the phosphorus goes in. And when this happens, the
phosphorus atom forms bonds with the surrounding silicon atoms. But now, because this atom,
phosphorus, had five valence electrons while the silicon it replaced had four,
there’s an extra electron floating around, we could say. The phosphorus impurity we’ve added
in has a full and stable set of eight valence electrons, and therefore this plus-one
electron becomes a free electron, liberated to move about throughout the
lattice.
So, by doping our semiconductor
with this impurity, a pentavalent atom, phosphorous, we have added a free electron
per a phosphorus atom beyond what the lattice had before. And by doing this, we increase the
conductive capacity of the semiconductor. Now, at this point, let’s introduce
a few symbols to this discussion. What we’ve just done is taken an
originally electrically neutral atom of phosphorus and substituted it in for silicon
in our atomic lattice. When we did that, one of these
valence electrons in the phosphorus atom was liberated and became a free
electron. As a result, the phosphorus atom
that remains has one fewer electron than it began with. And since it started out as
electrically neutral, having lost one negative charge through the release of this
electron, it’s now overall positive.
For this reason, it’s called a
positive donor ion. And the concentration of these
positive donor ions is given by N sub D with a superscript plus. An atom of phosphorus within a
lattice of silicon, as we’ve seen, becomes a positive ion through the loss of one of
its valence electrons. And because it gives this electron
to the lattice, we call it a donor. So, here the concentration of these
phosphorus atoms in our lattice and, in general, the concentration of any
pentavalent impurity we would add to a semiconductor is represented by N sub D
plus. And that’s to represent the
concentration of our positive donor ion.
Now, one important point about this
electron that is released by the phosphorus when the phosphorus joins into the
lattice, this electron, because it was an extra, we could say, beyond the eight that
fill up phosphorus’s outermost shell, does not leave a vacancy or a hole behind it
when it goes. Unlike in our pure semiconductor
sample where every free electron created a hole, here we have a free electron that
does not leave a hole behind. And that’s because it leaves behind
a phosphorus atom with a filled valence shell.
So, generally speaking, if we see a
free electron in a semiconductor, we know that it either came from a positive donor
ion, like phosphorus in this case, or it might have come from a broken covalent bond
between adjacent tetravalent atoms. In this case, we’re using
silicon. So, if we say that the density of
free electrons in our sample is lowercase 𝑛 and if we say that the density of our
electron holes is represented by lowercase 𝑝, where this letter is chosen because
these holes have an effective positive charge, then in a pure, that is, undoped,
semiconductor sample, 𝑛 is always equal to 𝑝 because every free electron released
leaves a hole behind.
But now that we’ve added an
impurity, which is able to release a free electron without leaving a hole, we would
say that in our doped semiconductor, it’s no longer the case that 𝑛 is equal to
𝑝. Rather, we would say that the free
electron density in our doped sample is equal to the density of holes, we can say
that this corresponds to the number of free electrons created by our tetravalent
atoms or silicon, plus the concentration of the positive donor ion that we’ve doped
our sample with. In this case, that’s
phosphorus.
As we look at this equation, we can
tell that as soon as we have added this dopant — that is, as soon as our positive
donor ion concentration is above zero — then in that case, 𝑛, our free electron
density, will be greater than 𝑝, our hole density. When this is the case, we say that
we’re working with an n-type semiconductor. And that simply means one with a
greater concentration of free electrons than holes. As we saw, the way it got that way
was by adding an impurity that had one more valence electron than the atoms of our
original undoped semiconductor did. This means that any time we have a
pure semiconductor sample made of a tetravalent atom, such as silicon, then if we
dope that sample with a pentavalent atom — such as, in this case, phosphorus — then
we are creating an n-type semiconductor.
So that’s one of the two main
semiconductor classes. And we can learn about the other
type by imagining replacing this central atom here, which used to be silicon and is
now phosphorus, with a trivalent atom. And our example of that is
boron. Just to see more clearly how this
works, let’s imagine we go back to having a lattice simply of silicon. So here we have this interior
silicon atom with a complete valence shell. And we’re gonna replace it with
this atom of boron, which we can see has three valence electrons and is also
electrically neutral. Meaning that as it is, it has no
net charge. When we make this substitution, see
what happens.
Our boron atom, because it shares
four valence electrons with surrounding silicon atoms, now has four plus three or
seven electrons in its outermost shell. Since eight valence electrons would
give the boron a complete set, we can say that there is a hole in its electron
orbitals. The presence of that hole means
that this atom of boron is very likely to accept a free electron from somewhere else
in the lattice. And in a similar way to how we
called our phosphorus dopant a positive donor ion, when our boron atom accepts a
free electron, it can fittingly be called a negative acceptor ion.
The reason we say it’s negative is
because before the boron was inserted into our semiconductor, it had no net
charge. But now that it’s accepted an
electron, which has a negative charge, our boron atom overall becomes a negative
ion. The symbol often used to represent
the concentration of these negative acceptor ions is N sub A minus. The A is because we’re talking
about a free electron acceptor. And then the minus sign in the
superscript reminds us that our dopant becomes a negative ion when it accepts such
an electron.
The fact that we can add impurities
to our semiconductor that increase the number of holes in it means that if at some
point we see a whole within the lattice of our semiconductor material, then we can
say it came from one of two sources. It either is a hole left behind by
the release of a free electron from our tetravalent silicon. Or, like this hole, it’s added in
through the addition of an impurity, a dopant. So, recalling that the density of
holes in our semiconductor overall is represented by lowercase 𝑝, we can say that
that density is equal to the density of free electrons, these would be electrons
released from our silicon atoms, plus the concentration of any negative acceptor
ions, that is, impurities.
We can see from this equation that
as soon as we dope our semiconductor with a trivalent atom like boron, then we will
contribute negative acceptor ions. And the density of positively
charged holes will exceed the density of negative free electrons. For that reason, the name given to
this kind of semiconductor is p-type. And this name just means it has a
greater density of positively charged holes than negatively charged free
electrons. So p-type and n-type semiconductors
are the two main classes, and we now see how to create these types of semiconductors
through doping.
What we’re seeing then is that when
we have a doped semiconductor, 𝑛 does not need to equal 𝑝. However, in a pure or undoped
sample, these two concentrations are equal. Say that we do have an undoped
semiconductor, that is, a pure semiconductor sample. We can say that in that sample, 𝑛
sub i represents the concentration of free electrons and the concentration of
holes. That means that if we square this
term, we’re effectively finding the density of free electrons times the density of
holes. And this 𝑛 i squared, which we got
from a pure or undoped semiconductor sample, it turns out is equal to 𝑛 times 𝑝,
where 𝑛 is given by this equation and 𝑝 is given by this one.
The overall meaning of this
equation is that in a pure semiconductor sample, if 𝑛 goes up, say, then 𝑝 must go
down. Or on the other hand, if 𝑝
increases, then 𝑛 must correspondingly decrease. Note that this is true for an
undoped sample, but not for a doped one. Now, one last observation we can
make is that when it comes to n-type and p-type semiconductors, it’s often the case
that, in the case of n-type, the positive donor ion concentration and, in the case
of p-type semiconductors, the negative acceptor ion concentration is much greater
than the concentration of holes and free electrons, respectively. When this is the case, it means the
effects of doping our semiconductor vastly outweigh the effects of raising the
temperature.
And if that’s so, we can make
approximations to these two equations. We can say, in the first case, that
𝑝 plus N sub D plus is approximately equal to just N sub D plus. This is saying that the great
majority of free electrons in our semiconductor are due to doping. And likewise, for our p-type
semiconductors, it’s often reasonable to approximate the hole concentration simply
as being equal to the negative acceptor ion concentration. Whether we can make these
approximations or not depends on the specifics of our semiconductor sample. But oftentimes, they are valid and
helpful to make.
Let’s now summarize what we’ve
learned about doped semiconductors. In this lesson, we saw that the
conductivity of a pure semiconductor can be varied by changing its temperature or by
doping. Doping consists of adding
impurities to the semiconductor lattice. These would be atoms with five or
three valence electrons. And lastly, we learned that there
are two types or classes of doped semiconductors. One is called n-type, where the
concentration of free electrons in the semiconductor is equal to the concentration
of holes plus the concentration of positive donor ions that come through doping.
And then for p-type, the
concentration of effectively positively charged holes in the lattice is equal to the
concentration of free electrons plus the concentration of negative acceptor
ions. The acceptor ions are part of the
lattice due, once again, to doping. In strongly doped semiconductors,
we can approximate the concentration of these donor and acceptor ions to be much
greater than the hole density or the free electron density, respectively, and
therefore approximately equal to the concentration of free electrons in an n-type
semiconductor or the concentration of holes in a p-type. This is a summary of doped
semiconductors.
