Subtitles section Play video Print subtitles [Slide 1] Ok, welcome to lecture 2. Now before we dive into the physics of transistors, what I want to do is to spend two lectures reviewing some basic concepts of semiconductors and semiconductor physics. Now, many of you have an extensive background in semiconductors and this will be pretty familiar to you. Some of you don't have much background in semiconductors and it's going to go pretty fast. Now what I mainly want to do is highlight the concepts that we're going to be using for the rest of the course. If you can get comfortable with using those concepts, you'll be set for the rest of the course. And the references give you some pointers to additional resources if you would like to fill in your gaps. So we have two parts of this lecture; Part One... [Slide 2] We'll just dive right into it and well go back and begin with basic freshman chemistry. So you'll remember that atoms have energy levels and silicon is an atom that has atomic number 14, so it has 14 electrons. Those 14 electrons have to fill in to these energy levels, n=1,2 ,3 ,4, etcetera. And we just start filling up the energy levels from the lowest energy until we have accounted for all fourteen electrons. And in order to do that, we end up filling in some of the n=3 energy levels, the two S levels are completely filled and then there are six states in the P level and we only need two of those and we have accounted for all 14 of the electrons that we need to. So deep down low energies, we call those the core levels, we don't need to worry much about them because there is not much we can do to affect them, but the highest energy levels are the ones that we worry about because we can manipulate those and they're involved in chemical bonding and those are the energy levels that we make use of in electronic devices. And the important point is that in the highest most energy levels we have four electrons, four valence electrons, though we have eight states there so there are four empty states as well. [Slide 3] Now, we are going to be primarily talking about transistors made on silicon. So think about a chunk of silicon. It has a lot of silicon atoms arranged in a regular lattice, 5 times 10 to the 22nd of them per cubic centimeter and they are arranged in this diamond lattice, each silicon atom has four nearest neighbors and the lattice spacing here is about five and a half Angstroms. Now something different happens when we put silicon in a lattice and it can bond with its nearest neighbors; those energy levels change and we going to need to discuss how they change. Important points to make are that we're only interested in the top most energy levels, the valence states, there are 8 of those. So that gives rise to 8N atoms states that we'll be interested in. But the interactions of the electrons wave functions as they interact with their neighbors changes the energy levels and that leads to what we call "Energy Bands." [Slide 4] So the energy levels become energy bands. The 3 S states and the 4 S states couple and merge and we end up with the same total number of states, we don't have simply 5 times 10 to the 22nd of these energy levels, they interact and the states become bands. We have half of the states end up creating a band of states where there are energy levels so finely spaced that we consider them to be continuous but 4N atoms states are in the lower band, 4N atoms states are in the upper band, all of the electrons then can be accommodated in the lower sets of states and there's a gap of energy in all of the states above are completely empty. We call that gap "The Forbidden Gap" because there are no states there. Electrons cannot be inside that gap. That's what happens at temperature equals zero. If we're about room temperature we have a little bit of thermal energy we can move an electron from a lower state to a higher state so we have a few empty states in the bottom band, the valence band, and we have a few electrons in the conduction band which is empty at T=0. [Slide 5] Ok so that allows us to explain what makes an insulator, what makes a semiconductor and what makes a metal. So an insulator is just a material that has a very big band gap. So consider silicon dioxide, for example, the insulator that is used as the gate insulator in most MOSFETs. It has a band gap of nine electron volts. The thermal energy is kT and that's roughly .026 electron volts so there isn't very much thermal energy and not nearly enough to break a bond and not enough to move an electron from the valence band to the conduction band so we don't have enough electrons to conduct electricity and the material is an insulator. Now a metal is completely different. In a metal it ends up that the states in one single band are filled only half way into the band so we have filled states and empty states and the electrons are now free to move if you apply a voltage, give them a little bit of energy they can just move up and fill one of the empty states and current will flow. Now a semiconductor is like an insulator, it just has a small band gap and its small enough that we can break a few of these bonds and at reasonable temperatures we can create some empty states in the valence band and we can create some electrons in the conduction band. That's a material that we would then call a semiconductor. The band gap of silicon is 1.1 electron volts. A band gap of a good insulator like silicon dioxide is 9 electron volts. [Slide 6] Now let's look a little more carefully within these bands of conduction band states and valence band states, we have all of the states distributed between a bottom energy and a top energy and then the valence band the top energy and a bottom energy. And they are distributed in some way throughout that energy range and we call that the density-of-states. It's the number of states per unit energy, typically, per unit volume. So there is one for the conduction band, one for the valence band. So these states are very finally spaced in energy, they came from those original atomic energy levels but they are smeared out and spaced so finely that we just consider this a continuous distribution of states. If I integrate over all of those energy ranges from the bottom to the top I'll just find the total number of states. So half of the states associated with that N atoms 5 times 10 to the 22nd atoms are located in the conduction band. The other half are located in the valence band. And it is important to realize that these states are extended in space; if an electron is in a silicon energy level of an isolated silicon atom, it's physically in a particular region not free to move throughout space. Inside a silicon crystal these states are extended which means these electrons are free to move throughout the crystal. The holes are actually free to move throughout the crystal also; we think of them as positive charge characters. Ok, now, we're only going to be able to perturb things, we are only going to be able to make a few empty states at the top of the valence band and a few filled states at the bottom of the conduction band so really all we're interested in is what happens near the edges of these bands. [Slide 7] And near the edges of these bands it turns out, for typical semiconductors, they have a simple shape; they tend to go parabolically with energy and then those of you who have had some semiconductor physics courses will have derived the density of states in this region near the bottom of the conduction band and you might remember that it goes as the square root of the energy with respect to the bottom of the band. And there are parameters called effective mass that come from that material's mass as well. We have a similar expression for the valence band; the number states in the valence band goes to the square root of the energy difference between the energy and the top of the valence band. So these are typical expressions that we will use. They assume common, simply energy bands that we call parabolic energy bands. But you should remember that the particular shape of the bands depends on details of the band structure and depends on whether or not the material is a 1D 2D or 3D material. [Slide 8] Ok, now I want to talk about something else we'll call an energy band diagram. We're going to spend a lot of time looking at energy band diagrams. And what do we mean by energy band diagram? So the electrons are "de-localized" and free to move. The holes are too. And that means everything is happening near the band edges where we know the densities of states. So it's very useful to plot the conduction band versus position and the valence band versus position and then we can get an intuitive feel for how the electrons and holes are moving throughout the silicon crystal. [Slide 9] Alright now, another important concept that you've seen before in semiconductor physics and we'll be making extensive use of is the Fermi function. So we know that typically in the valence band the states are mostly filled and in the conduction band they're mostly empty. If I were to make a plot of the probability that a state is occupied, the probability goes from zero to one and I know that the high energy states have a small probability and the low energy states have a high probability so I could just sketch what it should look like. And I know that way below the valence band the probability is one. Way above the conduction band bottom the probability is zero. And then makes a transition that is something like that. Now it turns out that there is a key parameter, we'll call the Fermi level. And the Fermi level is the energy at which the probability of a state being occupied is one half. Here I've drawn it in the forbidden gap, for which there are no states. If there were a state there, it would have probability one half to be occupied but there are no states there so there can be no electrons there. So there is a small probability of the states of the valence band of being empty because the probability is not quite one. And there is a small probability of the states at the bottom of the conduction band of being filled because the probability is not zero. And that particular function is well known, has a simple mathematical form called a Fermi function, and it simply gives us the probability that a state that a particular energy with respect to the Fermi energy is occupied. [Slide 10] So now we can sketch what we would call an n-type semiconductor. In an n-type semiconductor there are few electrons in the conduction band that are free to move. And in an n-type semiconductor that means that the Fermi level would be up closer to the conduction band such that there is some small probability that those states nearer to the conduction band will be occupied. [Slide 11] We could sketch a p-type semiconductor. So here is an energy band diagram; energy versus position for the valence band and the conduction band. We'll sketch the Fermi energy. We'll put it down valence band now because if I do that there's some small probability that the sates nearest the top of the