Quantum Dots PA2003: Nanoscale Frontiers Artificial atoms Schr ö dinger equation Square well...
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Quantum Dots
PA2003: Nanoscale Frontiers
• Artificial atoms• Schrödinger equation
• Square well potential• Harmonic oscillator• 2D Harmonic oscillator
• Real quantum dots• Semiconductors
• Semiconductor nanocrystals
Tipler Chapters 36,37
Quantum Dots
Dr Mervyn Roy, S6
Quantum Dots
PA2003: Nanoscale Frontiers
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
Artificial Atoms
Quantum Dots
PA2003: Nanoscale Frontiers
1D Standing waves
1 1
xx=0 x=L
V
Standing waves in a box
Quantum Dots
PA2003: Nanoscale Frontiers
1D Standing waves
1 1
xx=0 x=L
V
Standing waves in a box
Quantum Dots
PA2003: Nanoscale Frontiers
Schrödinger equation
Probability density
For stationary states
Uncertainty principleCan use to estimate energy, gives
Wave particle duality - probability waves described by the Schrödinger equation
Quantum Dots
PA2003: Nanoscale Frontiers
1D Square well confinement
1 1
xx=0 x=L
V
Same as standing waves in a box!
Discrete energy levels, quantum number nLowest energy state not zero!
Quantum Dots
PA2003: Nanoscale Frontiers
3D Square well confinement
a
cb
Because V(x,y,z) is separable (V=0) treat each direction separately
1 quantum number for each degree of freedom
• Squash box: energy level spacing in z very large, z motion quantised out -
effectively reduce the number of dimensions
• Stretch box: energy spacing very small - motion in y direction classical
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
Correspondence principle
Classical behaviour at high energywhen n is large
Shell fillingSpin up / down
1D quantum dot analogues of H, He etc.
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
source drainnanotube
SiO2dot
270 nm
gate 0.5 nm
• Nanotubes are already used in flak jackets, fuel pipes, tennis rackets etc.
• Molecular scale single electron transistor
2 electronchargedensity(Helium)
electrostatic confinement
potential
2 electrons per shell (spin up, spin down)
Quantum Dots
PA2003: Nanoscale Frontiers
Pillar dot
(20, 5/2)
vertical confinement ~ square welllateral confinement ~ 2D harmonic oscillator
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
Dots are highly strained
-0.10.0
5 n
m
InAs dot
GaAs
Isosurfaces in electron charge density
Quantum Dots
PA2003: Nanoscale Frontiers
Semiconductor bands
Eg
SemiconductorsElectrons: Holes:
Free particles:
Dispersion relations
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 depends on:• band gap Eg
Nanocrystals - photon depends on:• band gap Eg • nanocrystal size
small large
Quantum Dots
PA2003: Nanoscale Frontiers
Eg
Ee
Eh
Semiconductor nanocrystals
1 1
xx=0 x=L
V
~ 1D box,
Eg
Normal semiconductor
Semiconductor nanocrystals
Quantum Dots
PA2003: Nanoscale Frontiers
Semiconductor nanocrystals
Complications: 3D not 1D… R
Ee
Eh
makes no difference:
Complications: Electrons and holes present…
Quantum Dots
PA2003: Nanoscale Frontiers
Semiconductor nanocrystals
Complications 3D not 1D… R
Complications: Electrons and holes present…
Ee
Eh
makes no difference:
Coulomb interaction
Complications: surface effects, correlation effects etc. etc.
R
Quantum Dots
PA2003: Nanoscale Frontiers
Semiconductor nanocrystals
Gao et al. Nature Biotechnology, 22, (8), 969 (2004)