Quantum Dots PA2003: Nanoscale Frontiers Artificial atoms Schr ö dinger equation Square well potential Harmonic oscillator 2D Harmonic oscillator Real.

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)