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