1. Quantum Dots PA2003: Nanoscale Frontiers Artificial atoms Schr ö dinger 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

2. 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

3. Quantum Dots PA2003: Nanoscale Frontiers 1D Standing waves 11 x x=0x=L V Standing waves in a box

4. Quantum Dots PA2003: Nanoscale Frontiers 1D Standing waves 11 x x=0x=L V Standing waves in a box

5. Quantum Dots PA2003: Nanoscale Frontiers Schrödinger equation Probability density For stationary states Uncertainty principle Can use to estimate energy, gives Wave particle duality - probability waves described by the Schrödinger equation

6. Quantum Dots PA2003: Nanoscale Frontiers 1D Square well confinement 11 x x=0x=L V Same as standing waves in a box! Discrete energy levels, quantum number n Lowest energy state not zero!

7. Quantum Dots PA2003: Nanoscale Frontiers 3D Square well confinement a c b 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

8. Quantum Dots PA2003: Nanoscale Frontiers Harmonic confinement probability distributions

9. Quantum Dots PA2003: Nanoscale Frontiers Harmonic confinement probability distributions

10. Quantum Dots PA2003: Nanoscale Frontiers Harmonic confinement Correspondence principle Classical behaviour at high energy when n is large Shell filling Spin up / down 1D quantum dot analogues of H, He etc.

11. Quantum Dots PA2003: Nanoscale Frontiers 2D Harmonic confinement Solve Schrödinger equation in 2D StateEnergyquantum no’sspin no. e - total no. groundn=0, l =022 1stn=0, l = § 1 46 2ndn=1, l =0 or n=0, l = § 2 612

12. Quantum Dots PA2003: Nanoscale Frontiers Nanotube quantum dot source drain nanotube SiO 2 dot 270 nm gate 0.5 nm Nanotubes are already used in flak jackets, fuel pipes, tennis rackets etc. Molecular scale single electron transistor 2 electron charge density (Helium) electrostatic confinement potential 2 electrons per shell (spin up, spin down)

13. Quantum Dots PA2003: Nanoscale Frontiers Pillar dot (20, 5/2) vertical confinement ~ square well lateral confinement ~ 2D harmonic oscillator Electron molecule (pair correlation function) Rotating pentagonal electron molecule (Boron) Calculation by Prof. P. A. Maksym

14. Quantum Dots PA2003: Nanoscale Frontiers Self assembled quantum dot MBE grown dots. ~ 3D quantum box Dots are highly strained -0.10.0 5 nm InAs dot GaAs Isosurfaces in electron charge density

15. Quantum Dots PA2003: Nanoscale Frontiers Semiconductor bands EgEg Semiconductors Electrons: 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

16. Quantum Dots PA2003: Nanoscale Frontiers Semiconductor nanocrystals Bulk semiconductors – photon depends on: band gap E g Nanocrystals - photon depends on: band gap E g nanocrystal size small large

17. Quantum Dots PA2003: Nanoscale Frontiers EgEg EeEe EhEh Semiconductor nanocrystals 11 x x=0x=L V ~ 1D box, EgEg Normal semiconductor Semiconductor nanocrystals

18. Quantum Dots PA2003: Nanoscale Frontiers Semiconductor nanocrystals Complications: 3D not 1D… R EeEe EhEh makes no difference: Complications: Electrons and holes present…

19. Quantum Dots PA2003: Nanoscale Frontiers Semiconductor nanocrystals Complications 3D not 1D… R Complications: Electrons and holes present… EeEe EhEh makes no difference: Coulomb interaction Complications: surface effects, correlation effects etc. etc. R

20. Quantum Dots PA2003: Nanoscale Frontiers Semiconductor nanocrystals Gao et al. Nature Biotechnology, 22, (8), 969 (2004)