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  • [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