nanoHUB-U Nanoscale Transistors L1.2: The Transistor - Review of Semiconductors I

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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 valence band are not
occupied which will give me an empty state
or a hole that is free to move
around and act like a positive charge carrier.
[Slide 12] Now this Fermi level is very
interesting because I can now put the Fermi
level and the densities of states together and
I can relate them to the densities of
electrons or holes. So if I want to
know what is the density of electrons per
cubic centimeter in the conduction band, well it's
the number of states per cubic centimeter which
is that density of states in that energy
range times the width of that energy range
times the probability that the state is occupied,
which is given by the Fermi function, and
then I integrate from the bottom of the
conduction band. I should integrate from the top
but I'll take the integration all the way
to infinite because the Fermi function will make
the probability go to zero anyway so I'll
never get to the top. Well you just
plug in our expressions for the Fermi function
and for the densities of states and we
do the integration and we can find an
answer [Slide 13] So it takes a little
bit of math. Let me do it quickly
and you'll see how that works on one
of the homework problems for yourself. Here's the
Fermi function, here's the densities of states, and
this is the integral that we need to
do. I'll move all of the constants out
front so I have an integral like this
to do. I'll define some parameters. Eta is
normalized energy. It is the energy with respect
to the bottom of the conduction band in
units of KT. And Eta F is a
normalized Fermi energy. It's the Fermi energy with
respect to the bottom of the conduction band
in units of KT. And then I'll do
a change of variables and I'll convert my
integral to this form. Now I have an
integral that if I can work out we
have an answer. Ok. So I'll lump all
of the constants together and I'll call them
Nc. That is something that we call an
effective density of states. It involves things like
effective masses and kTs and is a material
dependent parameter. So that is something that is
known for any common semiconductor. We'll call it
the effective density of states as units per
number of states per cubic centimeter. [Slide 14]
Ok so if we do that integral, it
turns out you can't do that integral. You
can always do it numerically on a computer
so we'll just give it a name and
the name we give it is Fermi-Dirac integral
of order one half. The one half comes
because we have the normalized energy of the
one half power and we have a normalizing
fraction out here. It's just a name we
give to this particular function. The only way
we can evaluate that is to integrate it
with Simpson's rule or whatever numerically because there
is no analytical solution for that integral. But
that doesn't matter, we can evaluate it numerically.
So the electron density is related to the
effective densities of states which involves material dependent
parameters like effective masses. And it's related to
the position of the Fermi energy with respect
to the conduction band edge. So we have
the expression. [Slide 15] Now, Fermi-Dirac integrals make
things a little complicated mathematically. We're going to
try to avoid them for the most part
in this course but there will be a
time a or two where we will need
to use them. I just want to mention
something about Fermi-Dirac integrals and then point you
to some notes that you can find on
the nanohub that will tell you all that
you need to know about Fermi-Dirac integrals to
solve common semiconductor problems. This is a definition
of the general Fermi-Dirac integral. The normalizing factor
out front, the square root of pi over
2 that we saw is in general 1
over the gamma function of argument J plus
1. We dealt in the previous slide with
J of one half but in general J
could be an integer or any half integer.
So these are some useful parameters. The gamma
function, remember, is something like a factorial and
there's a recurrence relation that allows us to
determine it at one value if we know
it at a value that is just one
integer away. There are some nice properties such
that if we differentiate it we just kick
the order down one. And the Fermi-Dirac integrals
also simplify greatly. When this parameter, Eta F,
is much less than 1, much less than
0 actually, where the Fermi level is way
below the bottom of the conduction band then
the Fermi-Dirac integrals have any order just reduced
to an exponential. And that makes things simple
and that is the mostly the way that
we are going to be using them in
this course. Ok, now be sure when you're
dealing with Fermi-Dirac integrals that you don't confuse
this. When you're reading papers, you might come
across papers that talk about Fermi-Dirac integrals that's
defined a little differently and usually this will
written as a Roman F instead of as
a script F. The Roman F doesn't have
this normalization factor out front. And you have
to be careful about that. That normalization factor
is very nice, it allows us to differentiate
Fermi-Dirac integrals and use this expression but people
do it both ways and you have to
be careful to know how they're doing it.
[Slide 16] So if we make this assumption
that the Fermi level is way below the
bottom of the conduction band, then that Fermi-Dirac
integral simplifies to exponential and under those cases
we'll get some very simple expressions the Fermi-Dirac
integral becomes an exponential, Eta F is just
the Fermi energy minus the bottom of the
conduction band in units of kT. So we
get a very simple expression that just says
the electron density is exponentially related to the
distance to the bottom of the conduction band
and the Fermi energy. Higher the Fermi energy
the more the electron density is. So this
assumption would be called non-degenerate carrier statistics or
Boltzmann statistics, makes life simple, we will use
it frequently, every now and then we need
to be more accurate when we're analyzing experiments
then we'll direct to Fermi-Dirac integrals. [Slide 17]
Now you can do the same thing with
holes. The Fermi energy here is near the
conduction band so we so we have a
significant number of electrons in the conduction band
but there are a few holes and the
separation of the Fermi levels and the top
of the valence band determines how many. And
you can go through the same type of
argument and you can relate the hole density
to the Fermi energy with a Fermi-Dirac integral.
There is an effective density of states that
effects the hole effective mass and the parameter
Eta F now is the difference between the
top of the valence band and the Fermi
energy. [Slide 18] Ok. And I can simplify
the hole expression from Boltzmann statistics as well.
You know then if I look under equilibrium
conditions and if I take my simplified expression
for the electron density, my expression for the
whole density, and if I multiply the two
together I get this. And it turns out
then that the Fermi energies drop out, so
the result is independent of the Fermi energy
and the result just depends in the end
on the difference between the conduction band and
the valence band, that's the width of the
forbidden gap and on these two material dependent
parameters. Nc and Nv which depend on the
effective mass of electrons and holes. So the
product of N and P we'll call Ni
squared. And in an intrinsic semiconductor we have
an equal number of electrons and an equal
number of holes and that number is given
by this simple expression. In silicon it works
out beautifully. Room temperature in silicon, the intrinsic
density of electrons or holes is almost exactly
equal to 10 to the 10th per cubic
centimeter. That's a very small number. Remember there
are 5 times 10 to the 22nd silicon
atoms per cubic centimeter. [Slide 19] Ok, just
a couple more things and then we'll wrap
up. We can make semiconductor devices because of
doping. And that's what makes semiconductors so useful.
We can change the number of electrons in
the conduction band or the number of holes
in the valence band by controlling the location
of the Fermi energy and doing that by
doping. [Slide 20] And this is how it
works. Here is my silicon lattice, a little
cartoon picture. Each silicon atom has four valence
electrons and four nearest neighbors. So when you
put these all together with their four nearest
neighbors they have a completely filled shell, 8
electrons. And that is covalent bonding that makes
the silicon crystal. But if I substitute say
a phosphorus atom from column five of the
periodic table it has one extra electron. So
it goes into the lattice where it's supposed
to but there is one electron left over.
Or if I substitute a boron atom from
column three, it has one to few electrons.
So it goes in but there is a
missing bond in one place. Now if I
look at the electrons, it takes just a
little bit of energy to break that bond
and put that electron in the conduction band
and let it flow throughout the crystal. So
we would draw that energy of that energy
state of that fifth electron just below the
conduction band and at room temperature that electron
bond would be broken and the electrons would
be up here in the conduction band. We
would say that we've doped the semiconductor n-type.
Now if I look at this boron atom,
it takes just a little bit of energy
to break the next bond and put it
over here and fill that missing bond. Now
I've created a hole that's free to move
around through the semiconductor crystal, so I've doped
the semiconductor p-type. And that little energy I
would draw it down near the valence band.
So this is what we mean by n-type
and p-type doping. We dope n-type to make
the source and drain an n channel MOSFET.
We dope p-type to make the source and
drawn a p-channel MOSFET. So we have these
densities of dopants. At room temperature most of
those dopants are ionized. The fifth electron has
been released to the semiconductor crystal or in
the case of the boron atom, another electron
has filled that bond. [Slide 21] Ok and
then finally we can talk about how we
would in general determine the number of carriers
in the semiconductor. And we would do it
this way. A semiconductor likes to be neutral.
So if I look at the charge density
in coulombs per cubic centimeter, it's charge of
an electron times positive contribution from the holes
minus the negative contribution from the electrons plus
the positive contribution from the dopants so its
positively charged once that electron is broken off
minus the negatively charged acceptors. And electrons and
holes will fly around and try to make
everything neutral. So if I try to solve
that expression, if I assume that all the
dopants are ionized then I can assume this
is just the density of phosphorus atoms and
the density of boron atoms. And now I
know that product of N and P is
Ni squared, so I could eliminate P and
I will get a quadratic equation for N.
And you could do the algebra and what
you'll find is at near room temperature the
number of electrons in the conduction band is
very nearly the number of phosphorus atoms that
you put into the crystal. And the number
of holes in the valence band very nearly
the number of boron atoms that you put
in the semiconductor crystal. So it makes it
very easy for us to control the density
of the electrons in the conduction band and
holes in the valence band. [Slide 22] Ok,
so that's it for the first half of
this review of semiconductor fundamentals. The key points
I want to leave you with are the
importance of this density of states which describes
how the energy levels are distributed in a
conduction band and in the valence band, the
fact that the Fermi level is a critical
energy level that determines the probability that those
states are occupied, the fact that we can
mathematically relate the carrier densities to the location
of that Fermi level, and the fact that
what makes semiconductors so useful is that we
can control by doping the location of the
Fermi level and move it from anywhere down
near the valence band to up near the
conduction band edge. Ok so that's part one
of the review and we will continue the
review in the next lecture and then we
will be ready to dive into MOSFETs. Thanks
you.