Quantum Mechanics for

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• Quantum Mechanics for
• Electrical Engineers
• Dennis M. Sullivan, Ph.D.
• Department of Electrical and
• Computer Engineering
• University of Idaho

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Why quantum mechanics?

• At the beginning of the 20th century, various
  phenomena were being observed that could not be
  explained by classical mechanics.
• Energy is quantized
• Particles have a wave nature

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Source of electrons
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Source of electrons
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Conclusion
Particles have wave properties.
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Photoelectric Effect
Photoelectrons
Incident light
Material
The velocity of the escaping particles was
dependent on the wavelength of the light, not the
intensity as expected.
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Photoelectric Effect
Kinetic Energy T
Planck postulated in 1900 that thermal radiation
is emitted from a heated surface in discrete
packets called quanta.
Frequency f
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Einstein postulated that the energy of each
photon was related to the wave frequency.
h is Planks constant
This is the first major result energy is
related to frequency
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In 1924, Louis deBroglie postulated the existence
of matter waves. This lead to the famous
wave-particle duality principle. Specifically
that the momentum of a photon is given by
Or the more familiar form
Second major result Momentum is related to
wavelength
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Bottom line
Everything is at the same time a particle and a
wave.
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Physicists formulate everything as an energy
problem
Solve using only energy.
Problem Determine the velocity of the ball at
the bottom of the slope.
1 kg
1 meter
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While the ball is on the top of the hill, it has
potential energy.
Since the acceleration of gravity is
1 kg
the potential energy is
1 meter
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When it gets to the bottom of the hill, it no
longer has potential energy, but it has kinetic
energy. Since there have been no other external
forces, it must be the same
1 meter
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• There are several methods of advanced mechanics
  that change everything into energy.
• Lagrangian mechanics
• Hamiltonian mechanics

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• Erwin Schrödinger was taking this approach and
  developed the following equation to incorporate
  these new ideas
• This equation is 2nd order in time and 4th order
  is space.

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• Schrödinger realized that this was a completely
  intractable problem. (There were no computers in
  1921.) However, he saw that by considering y to
  be a complex function, he could factor above
  equation into two simpler equations, one of which
  is
• This is the Schrödinger equation.

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The parameter in the Schrödinger equation, y, is
a state variable. It is not directly associated
with any physical quantity itself, but all the
information can be extracted from it. Also,
remember that the Schrödinger equation was only
half of the real equation from which it was
derived. So to determine if a particle is
located between a and b, calculate
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This brings us to one of the basics
requirements of the state variable it must be
normalized
Note The amplitude of the state verctor y
does not represent the strength of the wave in
the usual sense. It is chosen to achieve
normalization
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Computer simulations can show how the Schrödinger
equation can model a particle like an electron
propagating as a wave packet.
The real part is blue, the imaginary part is red.
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• In quantum mechanics, physical properties are
• related to operators, call observables.
• Two of the fundamental operators are
• Momentum
• Kinetic energy

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The momentum operator (in one dimension) is
This seems pretty strange until I remember That
momentum is related to wavelenth
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We think of a wavepacket as a superposition of
plane waves
When I apply the momentum operator
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Kinetic energy can be derived from momentum
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If I want the expected value of the kinetic
Energy
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Note that smaller values of wavelength lead to
larger values of momentum and kinetic energy
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Lets look back at the Schrödinger equation
V(x) is the potential. It represents the
potential energy that a particle sees. Any
physical barrier must be modeled in terms of a
potential energy as seen by the particle.
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Conduction band diagram for n-type semiconductors
This is modeled in the SE as a change in
potential
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Similarly, if I want the expected value of the
Potential Energy
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Lets take a look at the Schrödinger equation and
some of the things we might be able to determine
from it.
Total energy Kinetic energy Potential
energy
The kinetic energy operator is
The potential energy operator is V
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Notice that as the wave interacts with the
potential, some kinetic energy is lost but some
potential energy is gained.
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The total energy stays the same!
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Now notice that part of the waveform has
continued propagating in the medium and part of
it has been reflected and is propagating the
other way.
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Does this mean that the particle has split into
two pieces?
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No! It means there is a probability it is
transmitted and a probablility is was reflected.
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Another example
Notice that the particle has KE of 0.127 eV, but
the barrier has potential energy of 0.2 eV.
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This is an important quantum mechanical phenomena
referred to as tunneling.
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• Fourier Transform Theory in Quantum Mechanics

Fourier theory is used in quantum mechanics, but
there are a couple differences.
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Instead of transforming from the time domain t to
the frequency domain , physicists
prefer to transform from the space domain x, to
the inverse space domain k.
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The other difference stems from the fact that the
state variable y is complex, so the FFT gives its
direction.
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Uncertainty or Where the _at_ am I?
Part of the mystique of quantum mechanics is the
famous Heisenberg Uncertainty principle.
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The uncertainty principle comes in two forms
1. You cant simultaneously know the position and
the momentum with unlimited accuracy.
2. You cant simultaneously know the time and the
energy with unlimited accuracy.
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We will start by looking at the first one
Remember that we said that momentum was related
to wavelength through Planks constant
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If we substitute this in for p and use the other
definition of Planks constant
we get
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Finally, we divide h from both sides leaving
This certainly took a lot of mystery out of it!
It says that I can only know position and
momentum to within a constant value.
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The uncertainty of quantum mechanics is like the
standard deviation of statistics.
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I can also apply this to the Fourier transformed
Y, which is the (1/l) domain
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All of the above signals and their Fourier
transforms were minimum uncertainty pairs because
I started with a Gaussian waveform in the space
domain. The Gaussian is the only signal that
Fourier transforms to the same shape.
Note that s is in the denominator in the space
domain and the numerator is the Fourier domain.
This is the mathematical expression of
uncertainty.
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This is the two-sided exponentially decaying
function. It is fairly close to minimum
uncertainty.
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Here is an extreme case of a sinusoid times a
rectangular function. The uncertainty if far
from minimum.
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Up until now, we have been talking about
propagating waveforms. However, we can also have
stationary solutions.
This is similar to an electromagnetic wave
confined to a cavity.
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One of the canonical problems in quantum
mechanics is the particle in an infinite well.
So we have to solve the Schrödinger equation in
the well subject to the boundary conditions
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We start with the Schrödinger equation
We are looking for a solution at the bottom of
the well where V0. Furthermore, we are looking
for a stationary solution, so there is no time
variation. And since this is a one-dimensional
problem,
Solutions are of the form
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Lastly, we have to find the magnitude constant A.
This is where the normalization comes in.
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Lets look again at the K parameter
Or solving for E
This means that the energies are quantized. The
energy of the electron is determined by the
order of the eigenfunction, not by the amplitude.
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If the base of the well is 100 A or 10 nm
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Only certain functions can exist in the
well. These are the Eigenfunctions.
Corresponding to the frequency of each
function is an energy given by
These are the eigenvalues, or Eigenenergies.
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In an infinite well, there will be an infinite
number of these eigenfunctions
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We start by initializing a particle in the
ground state, the lowest energy state, of the
10 nm well.
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As time progresses, the waveform oscillates
between the real and imaginary parts, but the
over-all magnitude stays the same.
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Note that the energy, 3.75 meV, is expressed as
kinetic energy, even though the particle is not
moving.
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As the simulation progresses, it will eventually
go back to its original state
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Is there a way we could have predicted this
time? Remember that energy is related to
frequency
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Here is at 60 meV
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Notice that the real and imaginary parts
oscillate much faster because it is at a much
higher energy.
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One of the most important principles in quantum
mechanics is the following If I have a
complete set of eigenfunctions for a system, I
can write any other function as a superposition
of these Eigenfunctions.
We recognize this as a Fourier series.
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Here are the first six eigenfunctions of the 10
nm infinite well.
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We have seen that if we initialize with an
eigenfunction, it will simple stay there.
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A propagating pulse will be made up of a
superposition of the eigenfunctions.
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As the pulse propagates, the magnitude of the
eigenfunction coefficients stays the same, but
the phases changes.
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This illustrates the importance of finding the
eigenfunctions f and the corresponding
eigenenergies En for a given system.
Because each eigenfunction evolves according to
its eigenenergy, I can predict how y(x,t) evolves.
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• Banding in
• Semiconductors

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Semiconductors are crystals. The atoms in a
semiconductor, e.g., silicon atoms are at very
specific locations.
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In many cases, the spacing of the atoms within
the crystal is more important that the atoms
themselves.
The periodic structure that indicates the
position of each atom is referred to as a lattice.
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Even though a semiconductor crystal is charge
neutral, there will be a variation in the
potential as seen by a free electron in the
crystal. For the sake of discussion, we will
simply model this as a small negative potential.
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We want to see what effect this lattice has on an
electron attempting to move in the semiconductor.
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Notice that some of the positive frequency has
been attenuated, but it appears as negative
frequency. This corresponds to a wave that is
propagating in the negative direction, i.e., has
been reflected.
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We might be inclined to ask the following
question If a well is other than infinite, will
it still have preferred states?
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The answer to that is yes!
In fact, even if the potential is negative the
spacing will want to retain certain states those
whose wavelength is a half-integer multiple of
the spacing.
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The answer to that is yes!
In fact, even negative potentials will attempt to
hold those states corresponding to their length.
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Therefore, the forbidden values of
i.e, any integer division of 2a is not allowed
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Notice that there is virtually no loss to the
positive frequency
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There is some loss, but it appears at l 20.
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Before we go any further, I wonder if we could
use Fourier transforms to help us predict which
wavelengths are forbidden.
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The potential of the lattice is described by
The Fourier transform of this function is
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This is almost the function we want. It gives a
spike every integer wavelength. To get one every
half-integer, we scale by a factor of ½.
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In the following, notice that the input wave
packet is narrower, leading to a broader pulse in
the frequency domain.
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If thats the case, what happens when I use a
very narrow pulse for the input?
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The next question I might ask is, What if I have
the same lattice spacing but a different
arrangement of potentials?
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These regions where the particle cant propagate
leads to the gaps in the E vs. k diagrams.
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We would like to use these concepts to think
about transport through a device, like a FET.
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Assume that the 3 wells are a very simple model
for a FET. If we can determine currents I1 and
I2, we know the current flow through the devices
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Start by initializing a particle in the left well
and see if it can flow to the right well.
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Obviously, it can flow.
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Heres another example.
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This one is clearly not flowing. Why?
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The 0.94 meV is an eigenstate of the 20 nm well.
However, the middle well is 10 nm and it doesnt
have an eigenstate at 0.94 meV. That is the
reason the particle cant flow it doesnt have
an eigenstate in the middle well!
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The previous waveform could go through the middle
well because 3.75 meV corresponds to the ground
state eigenfunction of the middle well.
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We saw earlier that 15 eV was an eigenenergy of
the 10 nm well, so we expect this one to go
through.
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Before too long, we can see the 2nd eigenfunction
start to form in the middle well.
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Eventually, it tunnels through.
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Suppose I were to go back and store the values at
the middle of the 10 nm well after I had
initialized it in one of the eigenstates. What
would I expect to see when I plotted the
resulting data?
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It would be an exponential with frequency
determined by
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I know that the Fourier transform of an
exponential is a delta function in the frequency
domain. In fact, each one of the eigenfunctions
results in a spike at a certain frequency, which
I can convert back to energy.
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This leads me to believe that I will only get
current flow if the particle is exactly at one of
the eigenenergies of the middle well.
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Of course, that represents the ideal situation
where we had an infinite 10 nm well. In
actuality, any given eigenfunctions is going to
decay out.
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Id like to know how fast particles decay in and
out of the channel. This will be the key to
determining transport through the channel.
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It is easier to measure the decay of the left
well into the middle well.
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So at one of the eigenenergies, the time
dependence is given by
This is a causal function, i.e, positive time.
Mathematically, wed rather deal with the
two-sided function
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The Fourier transform of the function
is
We can convert this to a function of energy
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We say that the eigenenergies have been
broadened.
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The broadening shows that even a waveform that is
not exactly at an eigenenergy of the middle well
has some probability of transmitting through.
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Consider the total current flow through the
channel
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Of course, if we had initialized the right well,
then
The total current is
The current flow depends on the following 1.
Available particles in the left or right well. 2.
Corresponding eigenstates in the channel. 3. The
escape rate corresponding to the energy
of the particle crossing the channel.
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Drain eigenenergies
Source eigenenergies
Channel eigenenergies
We might say if there is an eigenstate occupied
by the drain which is not occupied by the source,
and it corresponds to an allowed state in the
channel, we will get current flow.
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The more realistic situation is determined by the
Fermi-Dirac distribution
E is the energy of the state EF is the Fermi
energy, a parameter that is manipulated by
the applied voltage k is Boltzmanns constant T
is temperature in K
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Drain eigenenergies
Source eigenenergies
Now we have probabilities of the occupation of
states. This branch of physics is statistical
mechanics.
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The drain and source are usually modeled as
reservoirs. A reservoir may be thought of as an
infinite well. The consequence is, it can
provide or absorb a particle of any energy.
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Finding the Eigenfunctions of Arbitrary Structures
We have shown the importance of knowing the
eigenfunctions and corresponding eigenenergies of
a quantum structure. We made an analytic
calculation for an infinite well, but this is one
of the few quantum mechanics. MATLAB can
calculate them for one dimension.
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Let us start by assuming we dont know the
eigenfunctions of the 10 nm infinite well. I
will put in an impulse and see what happens.
(Actually, I use a fairly narrow Gaussian
pulse.) I will store the time-domain data at the
point of origin of the pulse.
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I am also taking the FFT of the time-domain data.
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Notice the two peaks in the Fourier domain at
3.75 meV and 15 meV. These correspond to the
first two eigenenergies of the 10 nm well.
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So now we know the eigenenergies en. How do we
find the eigenfunctions fn?
Any function, the input pulse included, must be a
superposition of the fn even if we dont yet know
them
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Lets look at just one point in the well, x0
Im going to take the discrete Fourier transform
at just one frequency corresponding to the ground
state energy
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The reason is the orthogonality of the
time-domain exponential functions.
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To get the entire eigenfunction f1 we have to
repeat this procedure at every point in the
problem space. It is not practical to store all
the time-domain data off-line and then perform
the DFT. But notice the following
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In other words, I could have a running DFT that
is calculating as the program is running.
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An arbitrary 3D structure
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Determination of Eigenenergies
Initialize the FDTD problem space with a test
function. As the simulation proceeds, record
data at the test point.
When the simulation is finished, Fourier
transform the recorded data.
Identify the eigenenergies from the peaks in the
Fourier transform.
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Eigenenergies for 3D Structure
Eigenenergies found at 552, 762,1150, 1530,
1830, 2162, 2560, 3405 meV
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Determination of the Eigenfunctions
Reinitialize the FDTD problem space with the
same test function.
As the program is running, calculate a discrete
Fourier transform at every cell in the problem
space at each eigenfrequency.
When the simulation is finished, use the data to
construct the eigenfunctions.
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First 4 Eigenfunctions
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Next 4 Eigenfunctions
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Vertical View of the Eigenfunctions
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Time Evolution (2162 meV)
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Time Evolution (contd)
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Observation
The mathematics presented in this talk is covered
in ECE 350.
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ECE Semiconductor Classes
ECE 460 Semiconductor Devices ECE 462/562
Semiconductor Theory ECE 465/565 Introduction
to Micorelectronics
Fabrication
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ECE 462/562 Semiconductor Theory
Spring 2010 MWF 1030-1120 am
This is primarily a crash course in quantum
mechanics as it pertains to semiconductors
specifically aimed at electrical engineers.
HW assignments are often simulation using simple
MATLAB programs.
The final exam is a presentation of a paper from
the literature that uses quantum semicondutor
theory.
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No one understands quantum mechanics
Richard Feynman
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