Almost There!

Interference and Review for 3rd Hour Exam

Review

• The probability of finding a particle in a particular region within a particular time interval is found by integrating the square of the wave function:

• P (x,t) = |(x,t)|2 dx = |(x)|2 dx• |(x)|2 dx is called the “probability density; the

area under a curve of probability density yields the probability the particle is in that region

• When a measurement is made, we say the wave function “collapses” to a point, and a particle is detected at some particular location

Particle in a box(x) = B sin (nx/a)

(x) |(x)|2n=2

n=3

Only certain wavelengths = 2a/n are allowedOnly certain momenta p = h/ = hn/2a are allowedOnly certain energies E = p2/2m = h2n2/8ma2 are

allowed - energy is QUANTIZEDAllowed energies depend on well width

“Real-World” Wells• Solution has non-trivial form, but only certain

states (integer n) are solutions• Each state has one allowed energy, so energy is

again quantized• Energy depends on well width a (confinement

width)

|(x)|2

n=1n=2

x

Quantum wells

• An electron is trapped since no empty energy states exist on either side of the well

Escaping quantum wells• Classically, an electron could gain thermal energy and

escape• For a deep well, this is not very probable. Given by

Boltzmann factor. TkEE BABe y Probabilit Relative

Escaping quantum wells• Thanks to quantum mechanics, an electron has a non-zero

probability of appearing outside of the well• This happens much more often than thermal escape if the

wells are close together.

Tunneling and Interference

• Can occur when total particle energy is less than barrier height.

• Particle can be scattered back even when its energy is greater than barrier height.

• What affects tunneling probability?T e–2kL

k = [82m(Epot – E)]½/h

A tunnel diode

• According to quantum physics, electrons could tunnel through to holes on the other side of the junction with comparable energy to the electron

• This happens fairly often• Applying a bias moves the

electrons out of the p-sideso more can tunnel in

The tunneling transistor

• As the potential difference increases, the energy levels on the positive side are lowered toward the electron’s energy

• Once the energy state in the well equals the electron’s energy, the electron can go through, and the current increases.

The tunneling transistor

• The current through the transistor increases as each successive energy level reaches the electron’s energy, then decreases as the energy level sinks below the electron’s energy

Quantum Entanglement(Quantum Computing)

• Consider photons going through beam splitters• NO way to predict whether photon will be

reflected or transmitted!

(Color of line is NOT related to actual color of laser; all beams have same wavelength!)

Randomness Revisited

• If particle/probabilistic theory correct, half the intensity always arrives in top detector, half in bottom

• BUT, can move mirror so no light in bottom!

(Color of line is NOT related to actual color of laser; all beams have same wavelength!)

Interference effects

• Laser light taking different paths interferes, causing zero intensity at bottom detector

• EVEN IF INTENSITY SO LOW THAT ONE PHOTON TRAVELS THROUGH AT A TIME

• What happens if I detect path with bomb?

No interference, even if bomb does not detonate!

Interpretation

• Wave theory does not explain why bomb detonates half the time

• Particle probability theory does not explain why changing position of mirrors affects detection

• Neither explains why presence of bomb destroys interference

• Quantum theory explains both!– Amplitudes, not probabilities add - interference– Measurement yields probability, not amplitude - bomb detonates

half the time– Once path determined, wavefunction reflects only that possibility -

presence of bomb destroys interference

Quantum Theory meets Bomb

• Four possible paths: RR and TT hit upper detector, TR and RT hit lower detector (R=reflected, T=transmitted)

• Classically, 4 equally-likely paths, so prob of each is 1/4, so prob at each detector is 1/4 + 1/4 = 1/2

• Quantum mechanically, square of amplitudes must each be 1/4 (prob for particular path), but amplitudes can be imaginary or complex!– e.g.,

TT22

1RR

22

1RT

2

1TR

2

1 ii

Adding amplitudes

• Lower detector:

• Upper detector:

TT22

1RR

22

1RT

2

1TR

2

1 ii

02

1

2

12

2

122

22

22

1

22

122

2

iii

What wave function would give 50% at each detector?

• Must have |a| = |b| = |c| = |d| = 1/4

• Need |a + b|2 = |c+d|2 = 1/2

TTRRRTTR dcba

TT22

RR22

RT22

1TR

22

1 ii

Pictorial Representation of 3D Integration Conceptusing Wafer Bonding,

* Figure adapted from IBM Corporation and used with permission.

Via Plug

Second Level(Thinned Substrate)

First Level

Third Level(Thinned Substrate)

Via Bridge

Bond

DeviceSurface

DeviceSurface

Bond(Face-to-face)

(Face-to-back)

DeviceSurface

Substrate

Substrate

Substrate

J. Lu et al

Broad band interconnect technology---high speed data transfer

Replacing electrical connection by optics:•Modulators/switches: electro-optic, optic-optic•Optical waveguides•Data compression (software)

Modulators guide

Chip stack

switches

fiber

Or: wireless!

light

Oriented & interconnected nanotube networks—Ajayan et al

– Local modification and Junction formation

– Termination (cutting of structures)

Catalyst

Junctions

Focused Ions

DNA and a little moreIvar Giaever

Rensselaer Polytechnic Instituteand

Applied BioPhysics, Inc.Troy, NY 12180

andOslo Universitetet

Blindern, Oslo

Wide Bandgap SemiconductorsWhat is a wide bandgap semiconductor?

Larger energy gap allows higher power and temperature operation and the generation of more energetic (i.e. blue) photons

The III-nitrides (AlN, GaN and InN), SiC have recently become feasible. Other materials (like diamond) are being investigated.

What are they good for?

How does a semiconductor laser work?

Stimulated vs. Spontaneous Emission (Cont.)

Derived in 1917 by Einstein. (Required for thermal equilibrium was it was recognized that photons were quantized.)

However, a “real” understanding of this was not achieved until the 1950’s.

Biased junction

n-type

p-type

depleted region(electric field)

Negativebias

photon out

MOSFET

• The potential difference between drain and source is continually applied

• When the gate potential difference is applied, current flows

Source Drain

n-type p-typen-type

Gate

(Metal-Oxide-Semiconductor, Field-Effect Transistor)

Einstein to the Rescue

• Einstein suggested that light was emitted or absorbed in particle-like quanta, called photons, of energy, E = hf

cresttrough

If an electron absorbs one of these photons, it gets the entire hf of energy.

If that energy is larger than the work function of the metal, the electron can leave; if not, it can’t:

Kmax = Eabs – = hf -

Bipolar Junction Transistor

n-type p-type n-type

Emitter BaseCollector

increasing electron energy

increasing hole energy

Bipolar Junction Transistor

http://hyperphysics.phy-astr.gsu.edu/hbase/solids/trans.html#c1

NOT Gate - the simplest case

Put an alternate path (output) before a switch.

If the switch is off, the current goes through the alternate path and is output.

If the switch is on, no current goes through the alternate path.

So the gate output is on if the switch is off and off if the switch is on.

OutputDump

Input

Switch

AND - slightly more complicated

AND gate returns a signal only if both of its two inputs are on.

Use the NAND output as input for NOT

If both inputs are on, the NOT input is off, so the AND output is on.

Else the NOT input is on, so the output is off.

Dump

Input Input

Switch Switch

Switch

Output