Transcript Class25_review - Rensselaer Polytechnic Institute

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) =  |Y(x,t)|2 dx =  |c(x)|2 dx
• |c(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
c(x) = B sin (npx/a)
n=3
c(x)
n=2
|c(x)|2
certain wavelengths l = 2a/n are allowed
 Only certain momenta p = h/l = hn/2a are allowed
 Only certain energies E = p2/2m = h2n2/8ma2 are
allowed - energy is QUANTIZED
 Allowed energies depend on well width
 Only
“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)
|c(x)|2
n=2
n=1
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.
EB  E A  k BT
Relative Probability  e
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 = [8p2m(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-side
so 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.,
1
1
1 i
1 i
Y  TR 
RT 
RR 
TT
2
2
2 2
2 2
Adding amplitudes
1
1
1 i
1 i
Y  TR 
RT 
RR 
TT
2
2
2 2
2 2
1 1
Y  
0
2 2
2
• Lower detector:
2
1 i 1 i
2  2i
• Upper detector: Y 


1
2 2 2 2
2 2
2
2
2
What wave function would give
50% at each detector?
Y  a TR  b RT  c RR  d TT
• Must have |a| = |b| = |c| = |d| = 1/4
• Need |a + b|2 = |c+d|2 = 1/2
Y
1
2 2
TR 
1
2 2
RT 
i
2 2
RR 
i
2 2
TT
J. Lu et al
Pictorial Representation of 3D Integration Concept
using Wafer Bonding,
Via Bridge
Via Plug
Substrate
Device
Surface
Third Level
(Thinned
Substrate)
Bond
(Face-to-back)
Substrate
Second Level
(Thinned
Substrate)
Device
Surface
Bond
(Face-to-face)
First Level
Device
Surface
Substrate
* Figure adapted from IBM Corporation and used with permission.
Broad band interconnect technology
---high speed data transfer
Or: wireless!
Replacing electrical connection by optics:
•Modulators/switches: electro-optic, optic-optic
•Optical waveguides
•Data compression (software)
Modulators guide
switches
light
fiber
Chip stack
Oriented & interconnected nanotube networks—Ajayan et al
Focused Ions
Catalyst
Junctions
– Local modification and Junction formation
– Termination (cutting of structures)
DNA and a little more
Ivar Giaever
Rensselaer Polytechnic Institute
and
Applied BioPhysics, Inc.
Troy, NY 12180
and
Oslo Universitetet
Blindern, Oslo
Wide Bandgap Semiconductors
What 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
Negative
bias
photon out
p-type
n-type
depleted region
(electric field)
MOSFET
(Metal-Oxide-Semiconductor, Field-Effect Transistor)
• The potential difference between drain and source is
continually applied
• When the gate potential difference is applied, current flows
Gate
Drain
Source
n-type
p-type
n-type
Einstein to the Rescue
• Einstein suggested that light was emitted or absorbed in
particle-like quanta, called photons, of energy, E = hf
If that energy is larger than
an electron
absorbs
theIfwork
function
of the one
of these
photons,can
it gets
metal,
the electron
leave;
if not,hf
it of
can’t:
the entire
energy.
Kmax = Eabs – F = hf - F
Emitter
Bipolar Junction
Transistor
Base
Collector
increasing
electron energy
increasing hole
energy
n-type
p-type
n-type
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.
Output
Input
Switch
Dump
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.
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
Output
Switch
Input
Switch
Input
Switch
Dump
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.