Lesson Video: Doped Semiconductors | Nagwa Lesson Video: Doped Semiconductors | Nagwa

Lesson Video: Doped Semiconductors Physics • Third Year of Secondary School

In this video, we will learn how to describe the effect that doping a semiconductor has on its electrical properties.


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.

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