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PA2003: Nanoscale Frontiers
Quantum Dots
• Artificial atoms
• Schrödinger equation
• Square well potential
• Harmonic oscillator
• 2D Harmonic oscillator
• Real quantum dots
• Semiconductors
• Semiconductor nanocrystals
Tipler Chapters 36,37
Dr Mervyn Roy, S6
Quantum Dots
PA2003: Nanoscale Frontiers
Artificial Atoms
Real atom: Electrons confined by coulomb
potential in 3D
- discrete energy levels
Quantum dot: any nanostructure that
confines electrons in 3D
- discrete energy levels
- much more flexibility than in nature
Applications: molecular scale electronics, spintronics, opto-electronics,
quantum cryptography, quantum computing, fluorescent bio-labels
Quantum Dots
PA2003: Nanoscale Frontiers
1D Standing waves
V
1
x=0
1
x=L
x
Standing waves in a box
Quantum Dots
PA2003: Nanoscale Frontiers
1D Standing waves
V
1
x=0
1
x=L
x
Standing waves in a box
Quantum Dots
PA2003: Nanoscale Frontiers
Schrödinger equation
Wave particle duality - probability waves described by the
Schrödinger equation
For stationary states
Probability density
Uncertainty principle
Can use to estimate energy, gives
Quantum Dots
PA2003: Nanoscale Frontiers
1D Square well confinement
V
1
x=0
1
x=L
x
Same as standing waves in a box!
Discrete energy levels, quantum number n
Lowest energy state not zero!
Quantum Dots
PA2003: Nanoscale Frontiers
3D Square well confinement
Because V(x,y,z) is separable (V=0) treat each direction separately
c 1 quantum number for each degree of freedom
b
a
• Stretch box:
energy spacing very small - motion in y direction classical
• Squash box:
energy level spacing in z very large, z motion quantised out -
effectively reduce the number of dimensions
10 % iso-surface
Quantum Dots
PA2003: Nanoscale Frontiers
Harmonic confinement
probability
distributions
Quantum Dots
PA2003: Nanoscale Frontiers
Harmonic confinement
probability
distributions
Quantum Dots
PA2003: Nanoscale Frontiers
Harmonic confinement
Shell filling
Spin up / down
1D quantum dot analogues of H, He etc.
Correspondence principle
Classical behaviour
at high energy
when n is large
Quantum Dots
PA2003: Nanoscale Frontiers
2D Harmonic confinement
Solve Schrödinger equation in 2D
State
Energy
quantum no’s
spin
no. e-
total no.
ground
n=0, l=0
2
2
1st
n=0, l=§1
4
6
2nd
n=1,l=0 or n=0,l=§2
6
12
Quantum Dots
PA2003: Nanoscale Frontiers
Nanotube quantum dot
• Nanotubes are already
used in flak jackets, fuel
pipes, tennis rackets etc.
nanotube
source
drain
270 nm
SiO2
gate
• Molecular scale single
electron transistor
dot
0.5 nm
electrostatic
confinement
potential
2 electrons per shell (spin up, spin down)
Quantum Dots
2 electron
charge
density
(Helium)
PA2003: Nanoscale Frontiers
Pillar dot
vertical confinement ~ square well
lateral confinement ~ 2D harmonic oscillator
(20, 5/2)
Electron molecule (pair correlation function)
Rotating pentagonal electron molecule (Boron)
Calculation by Prof. P. A. Maksym
Quantum Dots
PA2003: Nanoscale Frontiers
Self assembled quantum dot
MBE grown dots.
~ 3D quantum box
5 nm
InAs
dot
GaAs
0.0
-0.1
Dots are highly strained
Isosurfaces
in electron
charge
density
Quantum Dots
PA2003: Nanoscale Frontiers
Semiconductor bands
Dispersion relations
Free particles:
Eg
Semiconductors
Electrons:
Holes:
Hole (absence of electron): +ve charged particle with effective mass
holes and electrons recombine near k=0 to produce a photon
Quantum Dots
PA2003: Nanoscale Frontiers
Semiconductor nanocrystals
Bulk semiconductors – photon
• band gap Eg
Nanocrystals - photon
• band gap Eg
• nanocrystal size
small
Quantum Dots
large
depends on:
depends on:
PA2003: Nanoscale Frontiers
Semiconductor nanocrystals
Normal semiconductor
Eg
~ 1D box,
Semiconductor nanocrystals
V
1
Ee
1
Eg
Eh
x=0 x=L
Quantum Dots
x
PA2003: Nanoscale Frontiers
Semiconductor nanocrystals
Complications:
3D not 1D…
R
makes no difference:
Complications: Electrons and holes present…
Ee
Quantum Dots
Eh
PA2003: Nanoscale Frontiers
Semiconductor nanocrystals
Complications
3D not 1D…
R
R
makes no difference:
Complications: Electrons and holes present…
Ee
Eh
Complications: surface effects, correlation effects etc. etc.
Quantum Dots
Coulomb
interaction
PA2003: Nanoscale Frontiers
Semiconductor nanocrystals
Gao et al. Nature Biotechnology, 22, (8), 969 (2004)
Quantum Dots