Quantum Dots in Photonic Structures

Wednesdays, 17.00, SDT

Jan Suffczyński

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

Lecture 7: Low dimensional structures

Plan for today

1. Reminder

2. Doping and holes

3. Low dimensional structures

Wigner-Seitz Cell construction

Form connection to all neighbors and span a plane normal to the connecting line at half distance

Bloch waves

Felix Bloch1905, Zürich - 1983, Zürich

Bloch’s theorem:

Solutions of the Schrodinger equation

)()( rr rkk

ik ue

Periodic (unit cell) partEnvelope part

)()( Rrr kk uu

for the wave in periodic potential U(r) = U(r+R) are:

Bloch function:

)(Ψ )(Ψ)(V̂dr

d

2 2

22

rrr kkkm

Kittel

Nearly free electron model

Origin of a band gap!

Isolated Atoms

Diatomic Molecule

Four Closely Spaced Atoms

valence band

conduction band

Band formation

Electronic energy bands

allo

wed

ene

rgy

band

s

Brilluoin zones

m

kk 2

22

e (k): single parabola

folded parabola

0 /a-/a

E(k)

*2

22

m

k

Electron velocity and effective mass in the k-space

0 /a-/a

E(k)

/2a

-/2a

0 /a

-/a

v(k)

Velocity is zero at the top and bottom of energy band.

*2

22

m

k

Electron velocity and effective mass in the k-space

0 /a-/a

E(k)

/2a

-/2a

0 /a

-/a

v(k)

0

/a-/a

m*(k)

Velocity is zero at the top and bottom of energy band, the.

Efective mass: m*>0 at the band bottom, m*<0 at the band top, in the middle: m*→±∞ (effective mass description fails here).

*2

22

m

k

Electron velocity and effective mass in the k-space

Doping of semiconductors

Holes

• Consider an insulator (or semiconductor) with a few electrons excited from the valence band into the conduction band

• Apply an electric field– Now electrons in the valence band have some energy

states into which they can move– The movement is complicated since it involves ~ 1023

electrons

Holes

• We can “replace” electrons at the top of the band which have “negative” mass (and travel in opposite to the “normal” direction) by positively charged particles with a positive mass, and consider all phenomena using such particles

• Such particles are called Holes• Holes are usually heavier than electrons since they

depict collective behavior of many electrons

Low-dimensional structures

Dimensionality

2

Increase of the dimensionin one direction

2

2

2

Increase of the volume

21

22

23

1

B A B

z

B A B

z

z

A

quantum well quantum wire

quantum dot

Low-dimensional structures

Discrete States• Quantum confinement discrete states• Energy levels from solutions to Schrodinger Equation• Schrodinger equation:

• For 1D infinite potential well

• If confinement in only 1D (x), in the other 2 directions energy continuum

ErVm

)(2

22

integer n ,)sin(~)( Lxnx

mp

m

p

mLhn zy

228

22

2

22

Energy Total

x=0 x=L

V

Quantum WellsEnergy of the first confined level

Decrease of the level energy when width of the Quantum Well decreased

W. Tsang, E. Schubert, APL’1986

Quantum WellsEnergy of confined levels

GaAs/AlGaAs Quantum Well

R. Dingle, Festkorperprobleme’1975

In 3D…• For 3D infinite potential boxes

• Simple treatment considered– Potential barrier is not an infinite box

• Spherical confinement, harmonic oscillator (quadratic) potential

– Only a single electron• Multi-particle treatment• Electrons and holes

– Effective mass mismatch at boundary

integer qm,n, ,)sin()sin()sin(~),,( zyx Lzq

Lym

Lxnzyx

2

22

2

22

2

22

888levelsEnergy

zyx mL

hq

mLhm

mLhn

dE

dk

dk

dN

dE

dNDoS

V

k

kN

3

3

)2(

34

stateper vol

volspacek )(

Density of states

Structure Degree of Confineme

nt

Bulk Material 0D

Quantum Well

1D 1

Quantum Wire

2D

Quantum Dot 3D d (E)

dE

dN

E

E1/

Quantum Dots

QD asan

artificial atom

QD as an artificial atomAtom Quantum Dot

3D confinement of electrons

Discrete density of electron states

Emission spectrum composed of individual emission lines

Non-classical radiation statistics (e. g. single photon emission)

Creation of „molecules” possible

QD as an artificial atom- differences

Atom Quantum Dot

Size 0.1 nm 10 nm

Confining potential Coulombic (~1/r2) Parabolic

Electron binding energy

10 eV 100 meV

Interaction of electron with environement

Weak Strong (phonons, charges, nuclear spins…)

Anisotropy of confining potential

No Yes (shape, compoistion, strain…)

QD size

• Should be small enough to see quantum effect• kBT at 4.2 K ~0.36 meV --> for electron

maximum dimension in 1D ~100-200 nm

• Small size larger energy level separation

(Energy levels must be sufficiently separated to remain distinguishable under broadening, e.g. thermal)

QD types and fabrication methods

• Goal: to engineer potential energy barriers to confine electrons in 3 dimensions

• Basic types/methods– Colloidal chemistry– Electrostatic– Lithography– Epitaxy

• Fluctuation• Self-organized• Patterned growth

- „Defect” QDs

Colloidal Particles• 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 QDs

Red: bigger dots!

Blue: smaller dots!

Si nanocrystal, NREL

Electrostatically defined QDs

• Only one type of particles (electron or holes) confined--> (No spectroscopy)

• QW etching and overgrowth

Verma/NIST

• The advantage: QD shaping and positioning• The drawback: poor optical signal

(dislocations due to the etching!)

QW

Etching

Overgrowth

Lithography defined QDs

• Mismatch of bandgaps potential energy well

V. B. Verma et al., Opt. Express’2011

Lithography defined QDs

Lithography• Etch pillars in quantum well heterostructures

– Quantum well heterostructures give 1D confinement– Pillars provide confinement in the other 2 dimensions

• 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.

Flucutation type QDs

Flucutation of QW thickness

Flucutation of QW composition

Epitaxy: Self-Organized Growth

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, strain, – Control island initiation

• Induce local strain, grow on dislocation, vary growth conditions, combine with patterning

Lattice-mismatch inducedisland growth