Artificial atoms Schr ö dinger equation Square well potential Harmonic oscillator 2D Harmonic oscillator Real quantum

1. 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

2. 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

3. 1D Standing waves V 1 1 x x=0 x=L Standing waves in a box

4. 1D Standing waves V 1 1 x x=0 x=L Standing waves in a box

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

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

7. Squash box: energy level spacing in z very large, z motion quantised out - effectively reduce the number of dimensions 3D Square well confinement Because V(x,y,z) is separable (V=0) treat each direction separately 1 quantum number for each degree of freedom c b a • Stretch box: energy spacing very small - motion in y direction classical 10 % iso-surface

8. Harmonic confinement probability distributions

9. Harmonic confinement probability distributions

10. 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

11. State Energy quantum no’s spin total no. no. e- 2D Harmonic confinement Solve Schrödinger equation in 2D ground n=0, l=0 2 2 4 6 1st n=0, l=§1 2nd n=1,l=0 or n=0,l=§2 6 12

12. Nanotube quantum dot • Nanotubes are already used in flak jackets, fuel pipes, tennis rackets etc. • Molecular scale single electron transistor nanotube source drain 270 nm dot SiO2 0.5 nm gate 2 electron charge density (Helium) electrostatic confinement potential 2 electrons per shell (spin up, spin down)

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

14. InAs dot GaAs 5 nm Self assembled quantum dot MBE grown dots. ~ 3D quantum box 0.0 -0.1 Dots are highly strained Isosurfaces in electron charge density

15. Semiconductor bands Dispersion relations Free particles: Semiconductors Electrons: Holes: Eg Hole (absence of electron): +ve charged particle with effective mass holes and electrons recombine near k=0 to produce a photon

16. Semiconductor nanocrystals • Bulk semiconductors – photon depends on: • band gap Eg • Nanocrystals - photon depends on: • band gap Eg • nanocrystal size large small

17. V 1 1 x x=0 x=L Semiconductor nanocrystals Normal semiconductor Eg ~ 1D box, Semiconductor nanocrystals Ee Eg Eh

18. Ee Eh Semiconductor nanocrystals Complications: 3D not 1D… R makes no difference: Complications: Electrons and holes present…

19. Ee Eh Semiconductor nanocrystals Complications 3D not 1D… R R makes no difference: Complications: Electrons and holes present… Coulomb interaction Complications: surface effects, correlation effects etc. etc.

20. Semiconductor nanocrystals Gao et al. Nature Biotechnology, 22, (8), 969 (2004)