ec481 quantum dots-djb
TRANSCRIPT
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Quantum Dots
Understanding Quantum dots
(same material but size dependent bandgap size dependent emission)
Making things smaller (quantum)
Concepts and tools:
Particle-wave duality , Heisenberg's uncertainty relation Bohr model, De Broglie relation,
Wilson- Sommerfeld quantization rule, Schrdinger Eqn
First approximationSmaller size - quantization of energy Particle in a box (size assumed to be the whole story)
Refinement-influence of material
bandgap, effective mass, and exciton confinement
(bound e-h pair use the Bohr atom model)
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Quantum Confinement
3-D
All carriers act as free carriers in allthree directions
2-D or Quantum Wells
The carriers act as free carriers in a
plane First observed in semiconductor
systems
1-D or Quantum Wires
The carriers are free to move down thedirection of the wire
0-D or Quantum Dots
Systems in which carriers are confinedin all directions (no free carriers)
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From atoms to crystals
atom
cluster
Quantum DotSolid
Number of atom: 1 3-100
102-106 >106
Why the interest?
QD is an artificial atom - interesting physics!
High oscillator strength (interacts strongly with light)
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QD Applications(more later on applications )
Why the interest?
PV, LEDs, QD
dyes
Quantum optics applications
Quantum information
processing (QIP), single photon
sources ( reduce noise)
Flexible QD displays
sensitized solar cells,
multiple exciton generation,
intermediate band solar cells.
Fluorescence imaging
Photon anti-bunching Photo-voltaics
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3 primary methods Colloidal chemistry (bottom up)
Epitaxy (bottom-up, and bottom up with help)
Lithography (top-down)
Fabrication Methods
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Colloidal Particles (Bottom Up)
Engineer reactions to precipitate quantum dots from solutions or a host
material (e.g. polymer)
In some cases, need to cap the surface so the dot remains chemically
stable (i.e. bond other molecules on the surface)
Can form
core-shell
structures Typically group II-VI materials (e.g. CdS, CdSe)
Size variations ( size dispersion)
Evident Technologies: http://www.evidenttech.com/products/core_shell_evidots/overview.php
Sample papers: Steigerwald et al. Surface derivation and isolation of semiconductor cluster molecules. J. Am. Chem. Soc., 1988.
CdSe core with ZnS shell QDsRed: bigger dots!
Blue: smaller dots!
http://www.evidenttech.com/products/core_shell_evidots/overview.phphttp://www.evidenttech.com/products/core_shell_evidots/overview.php -
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Epitaxy: Self-Organized Growth (Bottom Up)
Strained Induced Self Assembly of
Quantum Dot Grow:
Lattice mis-match between InAs
and GaAs induces strain and results
in InAs islands on GaAs
GaAs
InAs (wetting layer)
GaAs
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Epitaxy: Self-Organized Growth
(Bottom Up) Self-organized QDs through epitaxial growth strains
Stranski-Krastanov growth mode (use MBE, MOCVD)
Islands formed on wetting layer due to lattice mismatch (size ~10s nm)
Disadvantage: size and shape fluctuations, ordering
Control island initiationInduce local strain, grow on dislocation, vary growth conditions, combine with patterning
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Fabrication of microposts
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Fabrication of microposts
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Epitaxy: Patterned Growth
(bottom up with help) Growth on patterned substrates
Grow QDs in pyramid-shaped
recesses
Recesses formed by selective ion
etching
Disadvantage: density of QDs
limited by mask pattern
T. Fukui et al. GaAs tetrahedral quantum dot structures fabricated using selective area metal
organic chemical vapor deposition. Appl. Phys. Lett. May, 1991
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Lithography (Top down)
Etch pillars in quantum well heterostructures
Quantum well heterostructures give 1D confinement
Mismatch of bandgaps potential energy well
Pillars provide confinement in the other 2 dimensions
Electron beam lithography Disadvantages: Slow, contamination, low density, defect formation
A. Scherer and H.G. Craighead. Fabrication of small laterally patterned multiple quantum wells. Appl. Phys. Lett., Nov 1986.
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Quantum Dot
Why the interest?
QD is an artificial atom - interesting physics!
High oscillator strength (interacts strongly with light)
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Atoms and light
Blackbody radiation from the sun has distinct absorption lines
Bohr model of the hydrogen atom - Quantized electron energy ,
quantized light energy
electrons allowed only in discrete energy levels, orbits
The orbits are given by a quantization condition of the
angular momentum L=pr=n
Excited atoms emit light at discrete energies
http://en.wikipedia.org/wiki/File:Emission_spectrum-H.png -
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Bohr model- hydrogen
Classical mechanics
Force balance:
Velocity :
Energy:
Classical description problem:
Positive nucleus circled by negative electrons accelerating charge should emit
radiation and loose energy, finally falling in to the nucleus (in ~ 10-6 s)
1885 1962
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Bohrs Postulates (1913)1) e- moves in circular motion about the nucleus under the influence of the Coulomb
interaction between the e- & nucleus, obeying the laws of classical mechanics
This postulate is based on the existence of the atomic nucleus
2) Instead of the infinity of orbits which would be possible in classical mechanics, e-
can only move in an orbit for which its orbital angular momentum L is an integer
multiple of hbar (h/2pi)
This postulate introduces quantization.
Angular quantization also leads to energy quantization.
3) E- moving in such an orbit does NOT radiate EM energy. Thus, its total energy E
remains constant.
This is based on stability
4) EM radiation is emitted if an e-
, initially moving in an orbit of total energy Ei,discontinuously changes its motion so that it moves in an orbit of total energy Ef. The
freq. of the emitted radiation is equal to the quantity =(Ei-Ef)/h This is basically linking to Einsteins postulate on energy-freq relation(1905)
THESE POSTULATES MIXES CLASSICAL & NON-CLASSICAL PHYSICS!!
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Bohr model- hydrogen
1) Force balance: Velocity : Angular momentum: L mvr
2) Quantization of angular momentum: L mvr n 22 2
2 2
0 02
0
2 2 2
0 2
0
14 4
4
144
n
e mv ne mv r m r
r r mr
n n er vme mr n
Potential Energy V at any finite distance (r) can be
obtained by integrating the work that would be done by
the Coulomb Force acting from r to inf:
2 2
2 2
0 04 4r
e eV dr
r r
Kinetic Energy K at any finite distance (r) can be obtained by:2
2
0
1
2 4 2
eK mv
r
Total Energy E=K+V=K-2K=-K2 4
2 2 2
0 0
1, 1,2,3,...
4 2 (4 ) 2
e meE n
r n
Quantization of the orbital angular momentum of the e- leads to a quantization of its
total energy.
n
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Bohr model- hydrogen
2 2
0 24n
nrme
Bohr radius when nucleus mass is infinite & electron in free space
For the first orbit, Bohr radius is r=5.3x10-11m ~ 0.53 Ao
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Bohr model- hydrogen
-- The normal state of the atom will be the state in which e- has the lowest energy, i.e., the
state n=1. This is called ground state (fundamental state)
-- When the atom receives energy (such as with an electric discharge or with high energy
photonic excitation), the electron can make a transition to a state with higher energy
(called excited state, with n>1).
-- Obeying Minimum energy tendency, the atom can emit its excess energy by EM
radiation and return to the ground state. The wavelength of the EM radiation is inversely
proportional to the energy difference.
4) EM radiation is emitted if an e-, initially moving in an orbit of total energy Ei, discontinuously
changes its motion so that it moves in an orbit of total energy Ef. The freq. of the emitted
radiation is equal to the quantity =(Ei-Ef)/h
24
2 3 2 2
0
1 1 1
(4 ) 4
i f
f i
E E mev
h n n
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Bohr model- hydrogen
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Bohr model- hydrogen
Binding energy of electron is
2 4
2 2 2
0 0
1 , 1,2,3,...4 2 (4 ) 2
n
n
e meE nr n
2
13.6, 1,2,3,...nE eV n
n
Correction 1) For finite size nuclear mass (M), replace m (free electron mass) by :
1*
1 /m
m M
Correction 2) If e- moves in medium, replace o by or
Binding energy scales by (/m)/(r2)
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Generalized Quantization Rule
In 1916, Wilson-Sommerfeld established a set of rules for the quantization of any physical
system for which the coordinates are periodic function of time.
Plank
s energy quantization & Bohr
s angular momentum quantization are special cases ofit.
For any physical system in which the coordinates are periodic function of time, there exists a
quantum condition for each coordinate. These quantum conditions are:
q qp dq n h
Where q is one of the coordinates, pq is the momentum associated with that coordinate, nq is a
quantum number which takes on integral values, and means that the integration is taken
ove one period of the coordinate q.
Let
s apply Wilson-Sommerfeld
s rule for Bohr
s angular momentum quantization.e- moving in circular orbit of radius r has an angular momentum, mvr=L, which is constant.
Angular coordinate is q, which is a periodic function of the time (qincreases from 0 to 2 inone period and this pattern repeats in each succeeding period).
q qp dq n h Ld nhq
2
0
2Ld L d L nh
q q
2nhL n
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Bohrs Postulates (1913)1) e- moves in circular motion about the nucleus under the influence of the Coulomb
interaction between the e- & nucleus, obeying the laws of classical mechanics
This postulate is based on the existence of the atomic nucleus
2) Instead of the infinity of orbits which would be possible in classical mechanics, e-
can only move in an orbit for which its orbital angular momentum L is an integer
multiple of hbar (h/2pi)
This postulate introduces quantization.
Angular quantization also leads to energy quantization.
3) E- moving in such an orbit does NOT radiate EM energy. Thus, its total energy E
remains constant.
This is based on stability
4) EM radiation is emitted if an e-
, initially moving in an orbit of total energy Ei,discontinuously changes its motion so that it moves in an orbit of total energy Ef. The
freq. of the emitted radiation is equal to the quantity =(Ei-Ef)/h This is basically linking to Einsteins postulate on energy-freq relation(1905)
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Interpretation of The Quantization Rule
q qp dq n h Ld nhq 2
02Ld L d L nh
q q
2nhL n
Physical Interpretation of Bohrs angular momentum quantization is given by de Broglie in 1924.
de Broglies hypothesis is based on wave-particle duality:
Just as a photon has a light wave associated with it, so a material particle (e.g. an e-) has an
associated matter wave that governs its motion.
Since universe is composed of matter & radiation, de Broglies suggestion is a statement
about a symmetry of nature.
1892-1987
Louis de Broglie
(1892 - 1987)
f h l
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Interpretation of The Quantization Rule
For matter and for radiation:
1) Total energy E of an entity is related to the frequency v of the wave by E=h2) Momentum p of the entity is related to the wavelength by p=h/lHere, the particle concepts (E & p) are connected through Plancks constant to thewave concepts ( & l)
2nhL n
2nhL mvr pr
2
h nhpr r
l
2 r n l
Combine Bohrs postulates (Sommerfeld Rule) with de Broglies postulate
Allowed orbits are those in which the circumference of the orbit can contain integer numbers
of de Broglie wavelengthStanding wave condition
l d l S f ld
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Generalized Quantization Rule
Lets apply Wilson-Sommerfelds rule for energy quantization in 1D Box.
Particles move between walls
q qp dq n h
0
0( ) ( ) 2
d
dmv dx mv dx dmv nh 2n
nhp mvd
2 2
28nn hE
md
18681951
Sommerfeld
http://en.wikipedia.org/wiki/File:Erwin_Schroedinger.jpg -
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Schrdinger equation (1D)
The wavefunction (x) a complex function (not a measurable number
The probability function P(x,t) dx=|(x,t)|2dx
Normalization |(x,t)|2dx=1
For V=0, traveling wave solution
Time-dependent SE
2 2
2( , ) ( , )( , ) ( , )
2x t x tV x t x t i
m x t
The time-independent 1D S.E:
2 2
2( ) ( ) ( ) ( )
2x V x x E x
m x
/( , ) ( ) iEtx t x e
The time-independent 1D S.E for ZERO POTENTIAL (i.e V=0):
2 2
2
( )( )2
xE xm x
1887 - 1961
http://en.wikipedia.org/wiki/File:Erwin_Schroedinger.jpg -
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Helmholtz eqn:
kSE=2m(E)/
022 SEk
2 2
2
( )( )
2
xE x
m x
kEM
= n/c
Schrdinger eqn:
c= c0 /n
Guess solution for electron!
Example:
For light in a Fabry Perot resonator with perfect reflection we had:
Eigenfunctions (modes)
standing wavesEigenvalues (quantization of allowed l)
Roundtrip length=n ln
Formal likeness of Helmholtz (light) and SE eqns (electron)
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Schrdinger equation (1D)
The wavefunction (x) a complex function (not a measurable number
The probability function P(x,t) dx=|(x,t)|2dx
Normalization |(x,t)|2dx=1
For V=0, traveling wave solution
Time-dependent SE
2 2
2( , ) ( , )( , ) ( , )
2x t x tV x t x t i
m x t
The time-independent 1D S.E:
2 2
2( ) ( ) ( ) ( )
2x V x x E x
m x
/( , ) ( ) iEtx t x e
The time-independent 1D S.E for ZERO POTENTIAL (i.e V=0):
2 2
2
( )( )2
xE xm x
2( ) ,ikx
mEx Ae where k
1887 - 1961
l b
http://en.wikipedia.org/wiki/File:Erwin_Schroedinger.jpg -
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Particle in a box
Stationary solution (Eigenstates)
The potential is zero in the well and infinite outside
Free particle (wave) inside the box
Boundary condition: The wave function goes to zero at the walls, (0) = (L) = 0
. integern,)sin(2
~)( L
xn
Lx
The energy increases quadratically with higher quantum number nThe energy increases ~ 1/L2
2 2
2
( )( ) ( ) ( )
2
xV x x E x
m x
l
http://en.wikipedia.org/wiki/File:Particle_in_a_box_wavefunctions.svghttp://en.wikipedia.org/wiki/File:Infinite_potential_well-en.svg -
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Scanning Tunneling Microscope
STM is based on the concept ofquantum tunneling.
When a conducting tip is brought very near to the surface to be examined, a bias (voltage difference) applied between
the two can allow electrons to tunnel through the vacuum between them.
The resulting tunneling currentis a function of tip position, applied voltage, and the local density of states (LDOS) of the
sample.
Information is acquired by monitoring the current as the tip's position scans across the surface, and is usually displayed
in image form.
STM image of the 2-atom
thick lead film.
Inset is a zoomed view
showing the atomic
structure.
(Dr. Ken Shih, UT Austin)
STM, invented in 1981 allows atomicresolution.
The microscope, for which two IBM
researchers Gerd Binnig and Heinrich
Rohrer received the 1986 Nobel Prize
in physics, revealed the topography of
surfaces, atom by atom.
E di l i f
http://en.wikipedia.org/wiki/Quantum_tunnelinghttp://en.wikipedia.org/wiki/Biasing_(electronics)http://en.wikipedia.org/wiki/Local_density_of_stateshttp://en.wikipedia.org/wiki/Local_density_of_stateshttp://en.wikipedia.org/wiki/Biasing_(electronics)http://en.wikipedia.org/wiki/Quantum_tunneling -
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Incidence angle is belowthe critical angle
Incidence angle is abovethe critical angle
No wave is transmitted
in region 2, but a little
bit leaks out. This is
the evanescent field Evanescent wav
TIR
Propagating wave
Evanescent wave perpendicular to interface
Micro-Ring Resonators
http://en.wikipedia.org/wiki/File:Evanescent_wave.jpghttp://en.wikipedia.org/wiki/File:Total_internal_reflection.jpghttp://en.wikipedia.org/wiki/File:Evanescent_wave.jpg -
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Micro-Ring Resonators
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Material Classification with Energy band Diagram
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Material Classification with Energy band Diagram
Ec=Conduction band
Ev=Valance band
Egap=Band Gap
Q t D t 1st d i ti f E i i E
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Quantum Dots-1st order approximation for Emission Energy
.
Quantization of electron & hole energy
, ,bg n h n hE E E E
bgE
Zero point energy for e-
Zero point energy for h+
Q t D t
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Quantum DotsL= 8 nm
.
2 2 22 2 2 2
2 2 2 6 2
( ) (1240)( ) 5
8 8 8*0.511 10 8
n
h hcE eV n n n meV n
mL mc L
Not the full story!
CdSe, 2 -8 nm
Check:
In bulk (for CdSe)Eg= 1.75 eV bulk emission l=708nm
In 2 nm dot, Energy increases by 2*94 meV (hole and electron confinement)
Eg = 175+0.188=1.938 eV
l=c/=hc/E hc 1240 (nm*eV)
l(nm) =1240/1.938 (eV)= 640 nm (still in the red)
This energy is not particularly large compared to the band gap, >1.5 eV
22
26
22
22
22
2
2
94210511.0*8
)1240(
8
)(
8)( nmeVnnLmc
hc
nmL
h
eVEn
Even if L=2 nm (very small dot)
Still a very small energy compared to the big shifts observed (red to blue)
, ,bg n h n hE E E E
CdS Q t d t
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Bawendi MIT
CdSe Quantum dots
Corrections optical excitation
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Corrections- optical excitation1) Electrons and holes in material will behave differently than in free space
Dielectric screening (band theory
Effective mass (band theory)Reduced mass (classical mechanics)
2 2
, * 22n e
e
nE
m R
me* and mh* are effective mass of electron and hole .
2 2
, * 22n h
h
nE
m R
bgE
2) Electrons & holes can bound to each other and form excitons
Their binding energy can be calculated using Bohr model!
3) Qdot is spherical spherical harmonics
22
, *2
nlnlm e
e
XE
m R
QDs Corrections
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QDs- Corrections
.
bgE
, ,( ) ( )bg exciton binding n h n hE E E E R E R
2
( / )13.6 o
n
r
mE eV
1) Zeroth level exciton binding energy:
22
1, *2
nl
e
e
X
E m R
22
1, *2
nlh
h
X
E m R
Where, Xnl=3.142
1) Zeroth level quantization energies:
CdSe, 2.3 -5.5 nm