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```Quantum Dots in Photonic Structures
Lecture 7: Low dimensional structures
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.
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
Bloch’s theorem:
Solutions of the Schrodinger equation
 2 d2 ˆ 
 
 V(r ) Ψ k (r )   k Ψ k (r )
2
 2m dr

for the wave in periodic potential U(r) = U(r+R) are:
Bloch function:
k (r )  e
i k r
uk (r )
Envelope part
Periodic (unit cell) part
uk (r )  uk (r  R )
Felix Bloch
1905, Zürich 1983, Zürich
Nearly free electron model
Origin of a band gap!
Kittel
Isolated Atoms
Diatomic Molecule
Four Closely Spaced Atoms
conduction band
valence band
Band formation
allowed energy bands
Electronic energy bands
Brilluoin zones
2
 k
k 
2m
2
 (k): single parabola
folded parabola
Electron velocity and effective mass in the k-space

2
2
 k

2 m*
Electron velocity and effective mass in the k-space

2
2
 k

2 m*
Velocity is zero at the top and bottom of energy band.
Electron velocity and effective mass in the k-space

2
2
 k

2 m*
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).
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
Increase of the
dimension
in one direction
Increase of
the volume
2
1
2
21
2
22
2
23
Low-dimensional structures
B
A
B
B
A
B
A
z
z
quantum well
z
quantum
wire
quantum dot
Discrete States
• Quantum confinement  discrete states
• Energy levels from solutions to Schrodinger Equation
• Schrodinger equation:

V
2

 2   V (r )  E
2m
• For 1D infinite potential well
( x) ~ sin( nLx ), n  integer
x=0
• If confinement in only 1D (x), in the other 2 directions  energy
continuum
T otalEnergy 
n2h2
8 mL2

p 2y
2m

p z2
2m
x=L
Quantum Wells
Energy of the first confined level
Decrease of the
level energy when
width of the
Quantum Well
decreased
W. Tsang, E. Schubert, APL’1986
Quantum Wells
Energy of confined levels
GaAs/AlGaAs Quantum Well
R. Dingle,
Festkorperprobleme’1975
In 3D…
• For 3D infinite potential boxes
 ( x, y, z ) ~ sin( nLxx ) sin( mLyy ) sin( qLzz ), n, m, q  integer
Energylevels 
n2h2
8 mL x 2

m2h2
8 mL y 2
q 2h2
 8mL 2
z
• 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
Density of states
DoS 
dN dN dk

dE dk dE
N (k ) 
k space vol
vol per state
4 3k 3

(2 )3 V
Structure
Degree of
Confinement
dN
dE
Bulk Material
0D
E
Quantum Well
1D
1
Quantum Wire
2D
1/ E
Quantum Dot
3D
d(E)
Quantum Dots
QD as
an
artificial atom
QD as an artificial atom
Atom
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
(Energy levels must be sufficiently separated to remain
distinguishable under broadening, e.g. thermal)
• Small size  larger energy level separation
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”)
CdSe core with ZnS shell QDs
Si nanocrystal, NREL
Red: bigger dots!
Blue: smaller dots!
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.
Electrostatically defined QDs
• Only one type of particles (electron or holes) confined
--> (No spectroscopy)
Lithography defined QDs
• QW etching and overgrowth
QW
Etching
Verma/NIST
•
Mismatch of bandgaps 
potential energy well
Overgrowth
• The advantage: QD shaping and positioning
• The drawback: poor optical signal
(dislocations due to the etching!)
Lithography defined QDs
V. B. Verma et al., Opt. Express’2011
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
Lattice-mismatch induced
island 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
```