EC481 Quantum Dots-DJB

7/27/2019 EC481 Quantum Dots-DJB

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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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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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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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-- 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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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


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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


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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
http://en.wikipedia.org/wiki/File:Infinite_potential_well-en.svg


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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

EC481 Quantum Dots-DJB

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