Transcript Quantum Dots in Photonic Structures

Quantum Dots in Photonic Structures
Lecture 5: Basics of Quantum mechanics and
introduction to semiconductors
Jan Suffczyński
Wednesdays, 17.00, SDT
Projekt Fizyka Plus nr POKL.04.01.02-00-034/11 współfinansowany przez Unię Europejską ze środków Europejskiego
Funduszu Społecznego w ramach Programu Operacyjnego Kapitał Ludzki
Plan for today
1.
Reminder
2.
Quantum
mechanics –
some insights
3.
Introdction
to semiconductors
Reminder. The Photonic Crystas from 1D to 3D
2D
1D
3D
Reminder. Bragg Diffraction
Wavelength corresponds to the
period.
Reflected waves are in phase.
Wavelength does not correspond to
the period
Reflected waves are not in phase.
Wave propagates through.
Wave does not propagate inside.
Yablonovitch, Sci.Am. 2001
The Bragg Mirror – A Basic PhC
10 Bragg pairs in air (nhigh=3.48, nlow=variable)
Normal incidence case
wavelength
The Bragg Mirror – A Basic PhC
10 Bragg pairs in air (nhigh=3.48, nlow=variable)
Normal incidence case
wavelength
Generalization of the Bragg mirror
Two-Dimensional Photonic Crystals
Purely 2D:
Deep-etched
Macroporous silicon
a = 1.5 mm
h = 100 mm
2D with vertical
confinement:
High vertical refractive
index contrast,
e.g. membranes
…or low vertical refractive
index contrast
e.g. GaAs or InP.
Energy gap in electromagnetic
spectrum
Increasing of the dielectric contrast could lead to the overlapping of energy gaps in any
direction in 3D space.
Photonic circuits
Florescu et al.
Intel
T-intersections and tight
bends, as in electric wiresnot posssible to achieve it in
dielectric waveguides.
Passive Building Blocks in PhC
Integrated Circuits
Cavities within Photonic Crystals
Introduce a defect into the periodic structure!
•
•
Creates an allowed photon state in the photonic band gap
 a cavity!
Why Quantum Physics?
• Classical Physics:
– provides successful description of every day, ordinary objects
• subfields: mechanics, thermodynamics, electrodynamics
Sir Isaac Newton
1. An object in motion tends to stay in motion.
2. Force equals mass times acceleration
3. For every action there is an equal and
opposite reaction.
Why Quantum Physics?
Quantum Physics:
 developed early 20th century, in response to fail of classical
physics in describing certain phenomena (blackbody
radiation, photoelectric effect, emission and absorption
spectra…)
• describes “small” objects (e.g. atoms and their constituents)
The Ultraviolet
Catastrophe
The Stern-Gerlach
Experiment
The Hydrogen
Spectrum
Fundamental postulates of the
quantum mechanics
Postulate 1: All information about a system is
provided by the system’s wavefunction
Postulate 2: The motion of a nonrelativistic particle is
governed by the Schrodinger equation
Postulate 3: Measurement of a system is associated
with a linear, Hermitian operator
Quantum Physics
• QP is “weird and counterintuitive”
• “Nobody feels perfectly comfortable with it “
(Murray Gell-Mann)
• “Those who are not shocked when they first come across
quantum theory cannot possibly have understood it”
(Niels Bohr)
• “I can safely say that nobody understands quantum
mechanics”
(Richard Feynman)
• But:
• the most successful physical theory in history
• underlies our understanding of atoms, molecules,
condensed matter, nuclei, elementary particles…
Einstein: quantum mechanics must be wrong
Quantum mechanics is either:
1. Incomplete
2. Incorrect
3. Or both
Quantum mechanics is certainly imposing. But an
inner voice tells me that it is not yet the real thing.
Quantum theory says a lot, but does not really bring
us any closer to the secret of the Old One. I, at any
rate, am convinced that He does not throw dice. - A.
Einstein
Quantum Mechanics: Real Black Magic Calculus. - A. Einstein
Postulate 1: All information about a system
is provided by the system’s wavefunction.
 ( x)
Pr( x )
x
x
1. The wavefunction can be positive, negative, or complex-valued.
2. The squared amplitude of the wavefunction at position y is
equal to the probability of observing the particle at position x.
𝜌 x dx = Ψ(𝑥, 𝑡) 2dx
∞
∞
ρ x dx =
−∞
Ψ(𝑥, 𝑡) 2 dx = 1
−∞
3. The wave function can change with time.
4. The existence of a wavefunction implies particle-wave duality.
Postulate 1: Particle wavefunction
Classical physics
100%
Quantum physics
99.99..%
1000000
-10
10
Quantum particles are usually delocalized, meaning they do not
have a well-specified position
Postulate 1: Uncertainty Principle
• It is impossible to measure simultaneously, with no uncertainty, the precise
values of k and x for the same particle. The wave number k may be
rewritten as
• For the case of a Gaussian wave packet we have
Thus for a single particle we have Heisenberg’s uncertainty principle:
Planck constant: h = 4.135667516(91)×10−15 eV*s = 6.62606957(29)×10−34 J*s
ℏ = h/2𝜋 = 6.58211928(15)×10−16 eV*s = 1.054571726(47)×10−34 J*s
Postulate 1: a particle can be put into a
superposition of multiple states at once
Ψ = Ψ1 + Ψ2 + Ψ3 + …
Classical ball:
Valid states:
Quantum ball:
Valid states:
White
White
Yellow
Yellow
+
White AND yellow
Electron Double-Slit Experiment
• The interference pattern
• The same behavior as for light!
C. Jönsson of Tübingen, Germany, 1961
The quantum mechanical explanation is that
each particle passes through both slits and
interferes with itself
The Quantum Explanation
Superposition
state
+
+
Detector

The wavefunction of each particle is a probability wave which
produces a probability interference pattern when it passes
through the two slits.
Postulate 2: The motion of a nonrelativistic
particle is governed by the Schrödinger
equation
Time-dependent S.E.:
Time-independent S.E.:


2
2
m
dx

2
Time-independent S.E.:
d
2

i
 (t )  Hˆ  (t )
t
Hˆ   E 

ˆ
 V ( x )   ( x)  E  ( x)

1. It is a wave equation whose solutions display superposition and
interference effects.
2. It implies that time evolution is reversible.
3. It is very difficult to solve for large systems (i.e. more than three
particles).
Postulate 2: A quantum mechanical particle can
tunnel through barriers rather than going over them.
Classical ball
Classical ball does not have
enough energy to climb hill.
Quantum ball
Quantum ball tunnels through
hill despite insufficient energy.
This effect is the basis for the scanning tunneling electron
microscope (STEM)
Postulate 2:
Quantum particles take all paths.
Classical picture
Quantum picture
Classical particles take a single
path specified by Newton’s
equations.
The Schrodinger equation
indicates that there is a
nonzero probability for a
particle to take any path
Postulate 3: Measurement of a quantum
mechanical system is associated with some
linear, Hermitian operator Ô.
Oˆ   Oˆ 
Oˆ   dx * ( x) Oˆ ( x)( x)
1. It implies that certain properties can only achieve a discrete set
of measured values
2. It implies that measurement is inherently probabilistic.
3. It implies that measurement necessarily alters the observed
system.
Postulate 3: Even if the exact wavefunction is
known, the outcome of measurement is
inherently probabilistic
Classical ball:
Quantum ball:
Before
measurement
+
or
After
measurement
For a known state, outcome
is deterministic.
For a known state, outcome
is probabilistic.
Postulate 3: Measurement necessarily
alters the observed system
Classical Elephant:
Before
measurement
Quantum Elephant:
+
After
measurement
State of the system is
unchanged by
measurement.
Measurement changes
the state of the system.
Postulate 3: Properties are actions to be
performed, not labels to be read
Classical Elephant:
Quantum Elephant:
Position = here
Color = grey
Size
= large
Position:
The ‘position’ of an object exists
independently of measurement
and is simply ‘read’ by the observer
‘Position’ is an action performed on
an object which produces some
particular result
In other words, properties like position or momentum do not
exist independent of measurement!
Wave Properties of Particles
• For photons:
E  h
E h h
p 

c
c

h
Or,  
p
• De Broglie hypothesized that particles of well defined
momentum also have a wavelength, as given above, the de
Broglie wavelength
De Broglie’s Hypothesis
• ALL material particles possess wave-like
properties, characterized by the wavelength λB,
related to the momentum p of the particle in the
same way as for light
de Broglie
wavelength of the
particle
Frequency:
E
f 
h
h
B 
p
Planck’s Constant
Momentum of the
particle
h
E  hf 
2f  
2
De Broglie’s Hypothesis
• De Broglie’s waves are not EM waves
– He called them “material” waves
– λB depends on the momentum and not on physical size of the
particle
h
h
h
B  

p mv
2Em
• For a non-relativistic free particle:
– Momentum is p = mv, here v is the speed of the particle
– For free particle total energy E, is kinetic energy
2
p
mv 2
EK

2m
2
“Construction” Particles From Waves
• Particles are localized in space
• Waves are extended in space.
• It is possible to build “localized” entities from a
superposition of number of waves with different values
of k-vector. For a continuum of waves, the superposition
is an integral over a continuum of waves with different kvectors.
– The wave then has a non-zero amplitude only within a limited
region of space
• Such wave is called “wave packet”
Wave Picture of Particle
•
•
Consider a wave packet made up of waves with a
distribution of wave vectors k, A(k), at time t.
The spatial distribution at a time t given by:

( x, t )   A(k ) cos(kx  t )dk
0
Probability of the Particle
•
The probability of observing the
particle between x and x + dx in
each state is
•
Note that E0 = 0 is not a possible
energy level.
•
The concept of energy levels, as
first discussed in the Bohr model,
has surfaced in a natural way by
using waves.
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Particle in a Box
•
•
•
•
A particle of mass m is trapped in a one-dimensional box of width L.
The particle is treated as a wave.
The box puts boundary conditions on the wave. The wave function must be zero at the
walls of the box and on the outside.
In order for the probability to vanish at the walls, we must have an integral number of
half wavelengths in the box.
•
The energy of the particle is
•
The possible wavelengths are quantized which yields the energy:
•
The possible energies of the particle are quantized.
.
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Crystal band structure
Crystal lattice
+
+
We define lattice points; these are points with identical environments
Crystal = lattice + basis
Crystal lattice
Crystal lattice
Crystal lattice
Choice of origin is arbitrary - lattice points need not be
atoms - but unit cell size should always be the same
Crystal lattice
Choice of origin is arbitrary - lattice points need not be atoms but unit cell size should always be the same
Crystal lattice
This is NOT a unit cell even though they are all the same - empty space is not allowed!
Crystal lattice – choice of the unit cell
no
yes