[L2.1 Slide 1] Well we're ready to start
Unit 2 on this course.
You recall that in unit one, we discussed
electrons and holes in semiconductors and
we thought of electrons and holes as
classical particles that behave according
to appropriately defined Newton's laws.
Well, quantum mechanics tells us
that a particle is also a wave,
a wave also behaves as a particle.
So we need some understanding of quantum
mechanics because it becomes important
to understand certain specific
features of semiconductor devices.
Some of you may have had a course
in quantum mechanics, and
this will be a refresher about some
key concepts that'll be important.
Some of you may have seen little or
no quantum mechanics previously, and
I hope that this will give
you enough of a feeling for
what the important considerations are.
We won't need a lot of quantum mechanics
in the remainder of the course.
We just want to understand the few key
concepts that we'll need as we move ahead.
[Slide 2] So, as I mentioned,
a free electron in vacuum,
not in a silicon crystal,
can be thought of as a particle.
We apply forces.
We acquire velocities and momentum.
We can compute the velocity and
momentum as a function of time
by applying Newton's laws.
Force is the driven of
momentum with respect to time.
This is classical Newtonian mechanics.
And we can compute how the momentum varies
as a function of time, how the velocity
varies as a function of time, and
how position varies as a function of time,
using equations that we were first
introduced to in freshman physics.
[Slide 3] Well, we need to go beyond this
classical physics to talk about how
electrons can behave sometimes like
particles and sometimes like waves,
and what the important considerations
are for semiconductors.
Now this unit is not a course
on quantum mechanics.
It's simply a review of some key concepts.
The need for quantum mechanics is
something that introductory courses
generally spend some time going through.
And this was all worked out in the early
part of the 20th century, as people began
to explain certain features of nature that
required going beyond classical physics.
[Slide 4] So the first one I'll mention
is black body radiation.
So if we take a body and we heat it up to
a temperature, it will emit radiation,
and a perfect black body
emitter will emit radiation
that has a certain characteristic of
intensity, or the amount of radiation,
the number of watts per square centimeter,
per unit frequency, that's coming off.
So the intensity versus wavelength
will have a characteristic that
looks like this.
There will be a peak intensity that
will come at a specific wavelength
that's about 2,900 micrometers
divided by the temperature in Kelvin.
No matter what material the body is made
of, at longer wavelengths, there will be
less radiation, and at shorter
wavelengths, there will be less radiation.
Now if we try to compute this
spectrum using classical physics,
there was a very big problem.
When you try to compute this spectrum,
it diverges.
There's something called
an ultraviolet catastrophe.
The intensity that you predict goes
to infinity as the wavelength gets
shorter and shorter.
And this is clearly wrong and
not in agreement with experiments.
And physicists struggled with this for
a long time trying to explain it.
In 1901, Max Planck discovered a way
to explain these characteristics.
He discovered that you could explain these
characteristics if you assumed that energy
was quantized into units
of Planck's constant,
this is a constant that he introduced,
times the frequency.
We could also write that as
Planck's constant divided by 2 pi
times 2 pi times the frequency.
We call Planck's constant,
divided by 2 pi h-bar.
And 2 pi times frequency and
cycles per second,
or hertz,
is the frequency in radians per second.
So, E is equal to h-bar omega,
or E is equal to hf, are two ways of
expressing this quantized energy.
It wasn't at all clear to Planck why the
energy should be quantized, but he could
explain the experimentally measured
characteristic if he assumed that it was.
[Slide 5] So a few years later,
Albert Einstein started thinking about how
do you explain the photoelectric effect.
The photoelectric effect occurs
when we shine light on a metal,
and then try to measure the number
of electrons that are ejected.
And what was found is that it didn't
matter how intense the light was,
how many photons per unit,
per second, per square centimeter.
They had to be above a certain
critical frequency before any
electrons were ejected from the solid.
And that critical energy was
the work function of the metal.
So Einstein showed that light could
be thought of as particles and
that each particle had an energy that was
the same quantized energy as Planck had
postulated, h times frequency, or
h-bar times omega, and when that energy
was higher than the work function of
the metal, electrons were ejected.
[Slide 6] About the same time,
people were also looking at the spectra,
the light given off by molecules or atoms,
so as you excite atoms into higher and
higher energetic states, and
then let then let them relax down,
they emit light, but
the light comes only in discrete colors,
not a continuous spectrum of colors.
Now Bohr was able to show that these
discrete energies can be explained,
if we treat the electrons as waves.
And that the frequencies that come off
will be given by the difference between
the energy levels,
the initial energy minus the final energy,
divided again by this Planck's constant or
h-bar.
So there's a variety of
evidence that's accumulating,
that particles behave like waves and
waves behave like particles.
[Slide 7] This is seen more clearly in
experiments of this kind.
Think of an electron gun that shoots
electrons between two holes in
an aperture.
And think about a screen here,
and we have a movable detector.
And when we move the screen,
the detector up and down, and
we detect the amount of electrons
that are coming through.
What we find is that we always detect
either no electron or a single electron.
We don't detect parts of electrons.
So this tells us that electrons
are behaving as discrete particles.
[Slide 8] However, if we do this experiment and
we do it many times and
we map out the distribution of intensity,
each time measuring one electron or
zero electrons and
then adding up all of the results,
the pattern that we get looks like the
kind of interference pattern that you see
when waves are interfering with each
other constructively and destructively.
So this tells us that
electrons behave like waves.
Okay, and Louis De Broglie in
1924 postulated that any particle
with a momentum, p, will behave
also like a wave and the relation
is that the momentum is equal to Planck's
constant divided by the wavelength,
and that allows us to go back and
forth between momentum and wavelength.
[Slide 9] So the summary of this experimental
evidence that was accumulating
was that energy is quantized, and
that waves can behave like particles,
and particles can behave like waves.
And a new theory that's to supplant
classical physics had to be developed
to explain these experimental findings.
[Slide 10] So we had, and
it was understood at that time that waves
can show the effects of quantization
when boundary conditions are applied.
And it was also understood that waves can
be localized to behave like particles
by adding up different wavelengths.
So this sort of suggested that
what we should search for
is a wave equation to describe particles.
[Slide 11] Well, that wave equation is shown here.
This is the famous Schrodinger wave
equation, and it's a function of time and
of position.
And this is not something you derive.
Just like you don't derive F = ma,
Newton's law.
That's simply a description of
a fundamental description of how nature
behaves, it doesn't come from
a more fundamental description,
at least not from
a semi-classical perspective.
Same way with the Schrodinger equation,
this is an equation that was discovered
that seems to explain how particles
behave when they behave like waves.
Well, how would we solve this equation?
Well, one technique to solve
a differential equation of this type
is to postulate that the time and
position dependents separate,
that the wave function, the solution
to this equation, is the product
of a function of position only, and
a product of a function of time only.
Now if you've had a course
in differential equations,
you may have worked through this
technique of separation of variables,
I won't do this now,
I'll simply tell you the result.
The result is that the time
dependent part of this equation
goes as e to the -i omega t.
Omega is related to energy by this
relation that we've seen before from
Planck and Einstein and others.
So the wave function itself
could also be written as
e to the i energy divided by h-bar, okay?
What about the space dependent part of it?
Well this technique of separation of
variables also allows us to solve for
the space dependent part of it.
[Slide 12] When we do that, we factor that time
dependent equation into a simpler equation
that only depends on space.
When we get done solving this equation and
we want the complete space and
time dependent wave function, we'll simply
multiply by its time dependent part,
e to the -i omega t.
So now all we have to do is to solve this
space dependent part of the wave equation.
The other postulate from quantum mechanics
then, is that the probability of
finding a particle at a particular
position in time is equal to psi star psi.
That's the probability of finding
a particle between x and x plus dx.
[Slide 13] All right, so let's see a little
bit more about how this works.
This is our time
independent wave equation,
just the part that describes
a special dependence.
Let me rearrange it a little bit, bring
the energy over to the left hand side and
write it like this.
If I look at this term here,
the solutions are gonna depend
on whether this term is positive or
whether it's negative.
So these are two separate cases.
[Slide 14] So if we look at this equation,
and if I consider first the case
where the energy of the electron is always
greater than this potential energy, U,
then I can lump everything together
here and call that quantity k squared.
And k squared is a positive quantity.
So I can write my wave
equation in this simpler form.
Now one way to solve an equation of this
form is simply to guess the solution,
plug it into the equation,
and show that it works.
If you guess that the solution is
an exponential, e to the i either plus or
minus kx, and
plug it into this equation, do the
differentiation, you'll see that it works.
It gives a solution to this equation.
The left hand side is equal to zero.
Okay, so if we then go back and ask for
the complete time dependence of this
solution, we take this special dependence,
we add in the e to the -i omega t for
the time dependence, and
we have our complete time and
space dependent wave function.
Now, I'll argue and we'll look at it
in a minute that this describes a wave
traveling in either the plus or
minus x direction.
[Slide 15] So let's look at that wave and
let's look at what velocity it travels at.
So here is my wave function,
e to the ikx minus omega t,
I'll look at the argument of that
exponential and I'll just call that theta,
and we'll assume that
the potential energy is constant.
That means k is constant.
And let's just follow a point of
constant phase in position, and
see what velocity that point moves at.
So we're gonna follow a position
of constant phase, so
the phase does not change with time.
The phase is kx minus omega t.
So we'll do that derivative, and we'll
get k times dx/dt, that's the velocity
at which the phase is propagating, minus
omega t, when we do the derivative there.
All right, so
I can solve that equation and
find dx/dt,
which we will call the phase velocity.
It's the velocity at which that
constant phase is propagating.
And you'll see that it's just given
by this very simple relation.
It's the frequency divided
by this wave number, k.
[Slide 16] So we have simple expression for
the velocity at which this wave travels.
We could also ask, what is the wavelength?
So if I look at this traveling wave,
I realize that at a given time,
if I move one wavelength in position, I'm
simply going to accumulate 2 pi of phase.
So if I look at position
x plus one wavelength,
that's going to be the same
phase as a position x.
But if I've undergone one wavelength,
I'll have accumulated
an additional phase of 2 pi.
Well, the phase is kx minus omega t
so at x plus one wavelength,
the phase is kx plus lambda
minus omega t.
At the previous point, it was kx minus
omega t, but it's 2 pi more than that.
So we solve this equation.
We find that k times
lambda is equal to 2 pi.
Solve that equation.
Remember, we know what k is,
k is simply related to the energy of
the electron that we're interested in, and
we know that, and to the potential energy,
U, and we presumably know that,
h-bar, here, we know that also.
So we know what k is.
So this allows us to
compute the wavelength.
It's simply 2 pi divided by k,
or we can turn it around and
say that the wave vector, k, which
we've encountered earlier in Unit 1,
is simply 2 pi divided by the wavelength.
Okay, so we can understand both the phase
velocity and the wavelength of these.
[Slide 17] What about the momentum?
We understand that the momentum of
a particle Is m times velocity.
What is the momentum of a wave?
Okay, well again,
we'll start with our same wave equation.
We'll solve it, we'll find that
the wave function is Ae to the ikx,
or one of the two solutions.
We know what k squared is, in terms
of the energy and potential energy.
So we can turn this around and
solve for the energy, and
we get energy is equal to potential
energy plus p squared over 2m,
p squared over 2m sounds
like kinetic energy.
So this says energy is equal to kinetic
energy divided by two times the mass.
Okay, well,
p squared here, I'm writing p
squared is h-bar k quantity squared.
So what this tells us is
that p is equal to h-bar k.
The momentum that we should
associate with this electron
is related to the wavelength
of the electron.
To this relation p is equal to h-bar, k that's what
Louis de Broglie postulated in 1924.
So it's all consistence with
the experimental observations that were
accumulated at that time.
[Slide 18] Okay so now let's apply this to a simple
example electrons in one dimension.
You know, 20, 30 years ago these used
to be thought of as simple mathematical
example problems that were unrealistic.
These days we can actually produce
semiconductor nanostructures that
are nanowires, where the electrons
behave as one dimensional particles.
So in this nanowire,
assuming a constant potential,
the solution would just be e to the i
plus or minus ikx depending on whether
the wave is propagating in the positive or
negative direction.
Well what if I take a chunk
of length between 0 and L and
say I have one electron here and
I'm trying to describe that one electron.
The probability of finding that
electron between 0 and L is
psi star psi dx.
If I integrate over the entire length,
that probability has to be 1 because I'm
assuming there's 1
electron in this nanowire.
And if we do that,
you can easily see that that allows us
to specify this arbitrary constant A.
As 1 over the square root of this
arbitrary normalization length.
So when we've done that, we've normalized
the wave function such that the integral
of psi star psi is 1, to describe the one
electron we're assuming that is here.
[Slide 19] Well we can also describe electrons in 2D.
So if I have a 2 dimensional
sheet in the x-y plane and
I have an electron in this
two dimensional sheet,
the wave function will be either the ikx
times x plus ky times y that's e to ik
parallel vector away vector
in the x-y plane time dotted with
role which is a position
vector in the x-y plane.
Now if I look at a portion of this x-y
plane with some cross sectional area A and
say there's one electron within
this region with area A,
I can normalize this wave function and
we'll find that the normalized
wave function brings down 1/this square root of a, okay?
[Slide 20] So, these are not textbook
mathematical examples anymore,
they're real materials, there are two
dimensional electronic materials,
graphene is only one example
of those kinds of materials.
We won't encounter these in the rest
of the course, but they exist, and
they're the subject of a great deal
of current interesting research.
[Slide 21] So finally, what about electron
waves in three dimensions?
So if I've got an electron in this
three dimensional volume omega,
the wave function will be e to
the i to kx times x plus ky times y
plus kz times kz, when I go through
the normalization process to describe
one electron in this volume, I'll find
that the arbitrary constant A is one
over the square root of volume, then we
have normalized the wave function in 3D.
[Slide 22] All right. So,
we spent some time to discuss
the solutions to the time independent wave
equation when the energy is
greater than the potential energy.
And we were doing it under the assumption
that the potential energy was constant.
What about the case where the energy
is less than the potential energy?
What do the wave functions
look like there?
[Slide 23] Well they'll look fundamentally different.
So we'll go back to our time
independent wave equation.
But in this case, the energy is
less than the potential energy, so
this term is negative.
So we'll define a quantity alpha squared,
which is U minus e
such that alpha squared
is a positive quantity.
And then we'll get a wave
equation of this form,
where remember alpha squared is a positive
quantity and we have a negative sign here.
Well if I ask for
what are the solutions to that equation,
again I can guess the solution.
If I guess that the solution is some
arbitrary constant A times e to the plus
or minus alpha x, I'll find that either
one of those satisfies this differential
equations, so the forms of the solutions
are e to the plus or minus alpha x.
This is a wave that is either
exponentially growing with position, or
exponentially decaying with position.
[Slide 24] So now we know what the general
feature of the wave function is
when its energy is less
than the potential energy.
When it's energy is greater
than the potential energy.
It behaves as a traveling wave.
When the energy is less that
behaves as a decaying or
as a growing exponential function.
That's a little bit about electron waves, but
remember, electrons are also particles.
[Slide 25] When we detect electrons,
we detect particles.
So, how do we describe electron particles?
So, let's think about an electron in
this nanowire somewhere between 0 and L.
We describe it by a wave function, but
if we ask what's the probability
of finding that electron?
We know that the electron is somewhere between
there but if I take psi star psi,
I'll find there's a uniform probability of finding
the electron anywhere between 0 and L.
Waves are everywhere, so
how do we describe particles?
[Slide 26] Well there's a well known
procedure that works for
any type of wave to localize the energy in
a wave to a particular spot in position.
And we can use that procedure
to localize electrons so
that they exist at one particular spot or
near one particular spot.
So let's try to describe an electron
that is located near x0.
The way to do that is to add up a bunch
of waves with different wavelengths and
add their amplitudes such that we put
together, if we use one amplitude
we know we have one wave vector,
the electron is spread out everywhere.
But if we have a number of
wave vectors spread out in
different amounts of amplitude,
then we can localize the electrons
such that the waves interfere
constructively near x0 but
they interfere destructively when we
get away from that particular location.
The wider the spread of wave vectors k,
the more we can localize this
at a particular position.
The narrower the spread, the wider
it becomes spread out in position.
So this way of forming of a packet of
waves with different wavelength is a way
to describe a particle that is located
near x equals x0 with a position
that is momentum is h-bar k,
which is near k0, which is
the center of this group of wave vectors
that we're using to localize the particle.
And there's a connection,
if we use a bigger spread of wave vectors,
we can localize the carriers to
a sharper and sharper in position.
And vice versa, so
the spread in k vector and
the spread in position is equal
to one-half, one can show.
[Slide 27] So these lead to some
uncertainty relations.
A wave packet that is very local in space,
is spread out in k-space.
A wave vector that is very localized in
k-space, is spread out in real space.
And that leads to some
uncertainty relations,
that apply to any wave, phenomenon.
And if we multiply delta k by h-bar
then we have the delta p for momentum.
So we get these very famous
uncertainty relations that between.
The uncertainty in momentum of an electron
and its uncertainty in position.
It has to be greater than or
equal to some minimum value, h-bar over 2.
And very similarly, if we have a wave
function that's sharply defined in time
then it's spread out in frequency.
If it's sharply defined in frequency
then it's spread out in time.
And there is an uncertainty principle
that relates those uncertainties.
The uncertainty in energy times
the uncertainty in time is greater than or
equal to h-bar over 2.
And as they propagate in time, the
uncertainty just grows bigger and bigger.
[Slide 28] So these are the famous uncertainty
relations that really describe
any wave phenomenon, but they're also
going to be describing the particles,
the quantum mechanical particles that
we describe with wave functions.
[Slide 29] All right,
just a few more concepts from quantum
mechanics that we need to understand here.
One is that if we take any wave,
we can describe a wave by its dispersion,
how its frequency varies with wave number.
Remember, the wave number is
2 pi over the wave length.
We can also work through an argument and
I won't do that here.
We've seen that the phase of each wave
propagates at the phase velocity, omega
divided by k, but a wave packet actually
propagates at a different velocity.
The center of the wave packet actually
propagates at a velocity that
is given by the slope of omega
with respect to k, d omega/dk.
So if we have free electrons,
remember a free electron is defined
by this relation E equals h-bar k squared
divided by 2 times the mass.
So if I ask what's the velocity
of an electron wave packet?
Centered near k is equal to k0.
The velocity of that wave
packet I would take d omega dk.
But energy is h-bar times omega,
another way of writing that
is 1 over h-bar times dE/dk.
So the velocity of an electron wave
packet is related to the slope of energy
versus momentum at that
particular location, okay?
So we can simply do that derivative for
this relation,
h-bar squared k squared over 2m and
find that the group velocity of this
electron wave is h-bar k divided by m.
That makes a lot of sense,
because h-bar k0 is
the momentum associated with
the wave packet itself.
Now, if you look at the phase velocity,
you can convince yourself that
this is not the phase velocity.
This wave packet is propagating
at a velocity that is
actually twice that of the phase velocity.
[Slide 30] Well, we've covered a lot of ground but
we're only gonna use a few key concepts
from quantum mechanics in this course and
I want to review the key concepts
from this particular lecture.
Classical mechanics describes,
The position and
momentum of particles with Newton's law,
F = ma.
In quantum mechanics,
it's a little more involved, but
we've discovered that
nature behaves differently.
We describe particles with a wave function
that has a space dependent part and
a time dependent part.
Energy and frequency are related
by a fundamental relation,
energy is h-bar times
frequency in radians per second or
Planck's constant times frequency and
cycles per second.
The space dependent part of
the wave equation we find by
solving the time independent
Schroedinger equation.
The probability of finding an electron
between x and x plus dx is proportional to
psi star psi
And then there are these
uncertainty relations.
The particles are actually
spread out a little bit.
They're spread out a little bit
in momentum and in position.
They're spread out
a little bit in energy and
in time according to these
uncertainty relations.
So these are a few key concepts.
We're going to explore how these play
out in some important considerations
relating to semiconductors and
we will do that.
We will begin doing that
in the next lecture.
Thank you.
