Loading...

PPT – Quantum Mechanics for PowerPoint presentation | free to download - id: 53c1ed-NDE5M

The Adobe Flash plugin is needed to view this content

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

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

Source of electrons

Source of electrons

Conclusion

Particles have wave properties.

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.

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

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

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

Bottom line

Everything is at the same time a particle and a

wave.

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

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

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

- There are several methods of advanced mechanics

that change everything into energy. - Lagrangian mechanics
- Hamiltonian mechanics

- 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.

- 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.

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

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

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.

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

- In quantum mechanics, physical properties are
- related to operators, call observables.
- Two of the fundamental operators are
- Momentum
- Kinetic energy

The momentum operator (in one dimension) is

This seems pretty strange until I remember That

momentum is related to wavelenth

We think of a wavepacket as a superposition of

plane waves

When I apply the momentum operator

Kinetic energy can be derived from momentum

If I want the expected value of the kinetic

Energy

(No Transcript)

(No Transcript)

Note that smaller values of wavelength lead to

larger values of momentum and kinetic energy

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.

Conduction band diagram for n-type semiconductors

This is modeled in the SE as a change in

potential

Similarly, if I want the expected value of the

Potential Energy

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

(No Transcript)

(No Transcript)

(No Transcript)

Notice that as the wave interacts with the

potential, some kinetic energy is lost but some

potential energy is gained.

(No Transcript)

(No Transcript)

The total energy stays the same!

(No Transcript)

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.

Does this mean that the particle has split into

two pieces?

No! It means there is a probability it is

transmitted and a probablility is was reflected.

Another example

Notice that the particle has KE of 0.127 eV, but

the barrier has potential energy of 0.2 eV.

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

This is an important quantum mechanical phenomena

referred to as tunneling.

- Fourier Transform Theory in Quantum Mechanics

Fourier theory is used in quantum mechanics, but

there are a couple differences.

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.

(No Transcript)

(No Transcript)

The other difference stems from the fact that the

state variable y is complex, so the FFT gives its

direction.

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

Uncertainty or Where the _at_ am I?

Part of the mystique of quantum mechanics is the

famous Heisenberg Uncertainty principle.

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.

We will start by looking at the first one

Remember that we said that momentum was related

to wavelength through Planks constant

If we substitute this in for p and use the other

definition of Planks constant

we get

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.

The uncertainty of quantum mechanics is like the

standard deviation of statistics.

I can also apply this to the Fourier transformed

Y, which is the (1/l) domain

(No Transcript)

(No Transcript)

(No Transcript)

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.

This is the two-sided exponentially decaying

function. It is fairly close to minimum

uncertainty.

Here is an extreme case of a sinusoid times a

rectangular function. The uncertainty if far

from minimum.

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.

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

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

Lastly, we have to find the magnitude constant A.

This is where the normalization comes in.

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.

If the base of the well is 100 A or 10 nm

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.

In an infinite well, there will be an infinite

number of these eigenfunctions

We start by initializing a particle in the

ground state, the lowest energy state, of the

10 nm well.

(No Transcript)

(No Transcript)

As time progresses, the waveform oscillates

between the real and imaginary parts, but the

over-all magnitude stays the same.

(No Transcript)

(No Transcript)

Note that the energy, 3.75 meV, is expressed as

kinetic energy, even though the particle is not

moving.

(No Transcript)

As the simulation progresses, it will eventually

go back to its original state

Is there a way we could have predicted this

time? Remember that energy is related to

frequency

(No Transcript)

Here is at 60 meV

(No Transcript)

(No Transcript)

(No Transcript)

Notice that the real and imaginary parts

oscillate much faster because it is at a much

higher energy.

(No Transcript)

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.

Here are the first six eigenfunctions of the 10

nm infinite well.

We have seen that if we initialize with an

eigenfunction, it will simple stay there.

(No Transcript)

(No Transcript)

A propagating pulse will be made up of a

superposition of the eigenfunctions.

As the pulse propagates, the magnitude of the

eigenfunction coefficients stays the same, but

the phases changes.

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

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.

- Banding in
- Semiconductors

Semiconductors are crystals. The atoms in a

semiconductor, e.g., silicon atoms are at very

specific locations.

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.

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.

We want to see what effect this lattice has on an

electron attempting to move in the semiconductor.

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

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.

We might be inclined to ask the following

question If a well is other than infinite, will

it still have preferred states?

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.

The answer to that is yes!

In fact, even negative potentials will attempt to

hold those states corresponding to their length.

Therefore, the forbidden values of

i.e, any integer division of 2a is not allowed

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

Notice that there is virtually no loss to the

positive frequency

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

There is some loss, but it appears at l 20.

Before we go any further, I wonder if we could

use Fourier transforms to help us predict which

wavelengths are forbidden.

The potential of the lattice is described by

The Fourier transform of this function is

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.

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

In the following, notice that the input wave

packet is narrower, leading to a broader pulse in

the frequency domain.

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

If thats the case, what happens when I use a

very narrow pulse for the input?

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

The next question I might ask is, What if I have

the same lattice spacing but a different

arrangement of potentials?

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

These regions where the particle cant propagate

leads to the gaps in the E vs. k diagrams.

We would like to use these concepts to think

about transport through a device, like a FET.

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

Start by initializing a particle in the left well

and see if it can flow to the right well.

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

Obviously, it can flow.

Heres another example.

(No Transcript)

(No Transcript)

This one is clearly not flowing. Why?

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!

The previous waveform could go through the middle

well because 3.75 meV corresponds to the ground

state eigenfunction of the middle well.

We saw earlier that 15 eV was an eigenenergy of

the 10 nm well, so we expect this one to go

through.

(No Transcript)

(No Transcript)

Before too long, we can see the 2nd eigenfunction

start to form in the middle well.

Eventually, it tunnels through.

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?

It would be an exponential with frequency

determined by

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.

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.

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.

(No Transcript)

(No Transcript)

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.

It is easier to measure the decay of the left

well into the middle well.

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

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

The Fourier transform of the function

is

We can convert this to a function of energy

(No Transcript)

We say that the eigenenergies have been

broadened.

The broadening shows that even a waveform that is

not exactly at an eigenenergy of the middle well

has some probability of transmitting through.

(No Transcript)

Consider the total current flow through the

channel

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.

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.

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

Drain eigenenergies

Source eigenenergies

Now we have probabilities of the occupation of

states. This branch of physics is statistical

mechanics.

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.

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.

(No Transcript)

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.

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

I am also taking the FFT of the time-domain data.

(No Transcript)

(No Transcript)

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.

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

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

The reason is the orthogonality of the

time-domain exponential functions.

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

In other words, I could have a running DFT that

is calculating as the program is running.

An arbitrary 3D structure

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.

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

Eigenenergies for 3D Structure

Eigenenergies found at 552, 762,1150, 1530,

1830, 2162, 2560, 3405 meV

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.

First 4 Eigenfunctions

Next 4 Eigenfunctions

Vertical View of the Eigenfunctions

Time Evolution (2162 meV)

Time Evolution (contd)

Observation

The mathematics presented in this talk is covered

in ECE 350.

ECE Semiconductor Classes

ECE 460 Semiconductor Devices ECE 462/562

Semiconductor Theory ECE 465/565 Introduction

to Micorelectronics

Fabrication

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.

No one understands quantum mechanics

Richard Feynman

(No Transcript)