Quantum Dots in Photonic Structures

1. Lecture 5: Basics of Quantum mechanics and introductionto semiconductors Quantum Dots in PhotonicStructures Wednesdays, 17.00, SDT Jan Suffczyński Projekt Fizyka Plus nr POKL.04.01.02-00-034/11 współfinansowany przez Unię Europejską ze środków Europejskiego Funduszu Społecznego w ramach Programu Operacyjnego Kapitał Ludzki

2. Plan for today Reminder 2. Quantum mechanics – someinsights 3. Introdction to semiconductors

3. 2D 1D 3D Reminder. The Photonic Crystas from 1D to 3D

4. Reminder. Bragg Diffraction Wavelength does not correspond to the period Reflected waves are not in phase. Wave propagates through. Wavelength corresponds to the period. Reflected waves are in phase. Wave does not propagate inside. Yablonovitch, Sci.Am. 2001

5. 10 Bragg pairs in air (nhigh=3.48, nlow=variable) Normal incidence case wavelength The Bragg Mirror – A Basic PhC

6. 10 Bragg pairs in air (nhigh=3.48, nlow=variable) Normal incidence case wavelength The Bragg Mirror – A Basic PhC Generalization of the Bragg mirror

7. Two-Dimensional Photonic Crystals Purely 2D: Deep-etched Macroporous silicon a = 1.5 mm h = 100 mm …or low vertical refractive index contrast e.g. GaAs or InP. High vertical refractive index contrast, e.g. membranes 2D with vertical confinement:

8. Energy gap in electromagnetic spectrum Increasing of the dielectric contrast could lead to the overlapping of energy gaps in any direction in 3D space.

9. Photonic circuits Florescu et al. Intel T-intersections and tight bends, as in electric wires- not posssible to achieve it in dielectric waveguides.

10. Passive Building Blocks in PhC Integrated Circuits

11. CavitieswithinPhotonicCrystals Introduce a defect into the periodic structure! • Creates an allowed photon state in the photonic band gap •  a cavity!

12. Why Quantum Physics? • Classical Physics: • provides successful description of every day, ordinary objects • subfields: mechanics, thermodynamics, electrodynamics 1. An object in motion tends to stay in motion. 2. Force equals mass times acceleration 3. For every action there is an equal and opposite reaction. Sir Isaac Newton

13. Why Quantum Physics? • Quantum Physics: • developed early 20th century, in response to fail of classical physics in describing certain phenomena (blackbody radiation, photoelectric effect, emission and absorption spectra…) • describes “small” objects (e.g. atoms and their constituents) The Stern-Gerlach Experiment The Hydrogen Spectrum The Ultraviolet Catastrophe

14. Fundamental postulates of the quantum mechanics Postulate 1: All information about a system is provided by the system’s wavefunction Postulate 2:The motion of a nonrelativistic particle is governed by the Schrodinger equation Postulate 3: Measurement of a system is associated with a linear, Hermitian operator

15. Quantum Physics • QP is “weird and counterintuitive” • “Nobody feels perfectly comfortable with it “ (Murray Gell-Mann) • “Those who are not shocked when they first come across quantum theory cannot possibly have understood it” (Niels Bohr) • “I can safely say that nobody understands quantum mechanics” (Richard Feynman) • But: • the most successful physical theory in history • underlies our understanding of atoms, molecules, condensed matter, nuclei, elementary particles…

16. Einstein: quantum mechanics must be wrong Quantum mechanics iseither: 1. Incomplete 2. Incorrect 3. Or both Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. Quantum theory says a lot, but does not really bring us any closer to the secret of the Old One. I, at any rate, am convinced that He does not throw dice. - A. Einstein Quantum Mechanics: Real Black Magic Calculus. - A. Einstein

17. Postulate 1: All information about a system is provided by the system’s wavefunction. x x 1. The wavefunction can be positive, negative, or complex-valued. 2. The squared amplitude of the wavefunction at position yis equal to the probability of observing the particle at position x. 3. The wave function can change with time. 4. The existence of a wavefunction implies particle-wave duality.

18. Postulate 1: Particle wavefunction Classical physics Quantum physics 100% 1000000 10-10 99.99..% Quantum particles are usually delocalized, meaning they do not have a well-specified position

19. Postulate 1: Uncertainty Principle • It is impossible to measure simultaneously, with no uncertainty, the precise values of k and x for the same particle. The wave number k may be rewritten as • For the case of a Gaussian wave packet we have Thus for a single particle we have Heisenberg’s uncertainty principle: Planck constant: h = 4.135667516(91)×10−15eV*s = 6.62606957(29)×10−34 J*s = 6.58211928(15)×10−16eV*s = 1.054571726(47)×10−34 J*s

20. Postulate 1: a particle can be put into a superposition of multiple states at once Classical ball: Quantum ball: Valid states: Valid states: White White Yellow Yellow + White AND yellow

21. The interference pattern The same behavior as for light! Electron Double-Slit Experiment C. Jönsson of Tübingen, Germany, 1961

22. Superposition state + + Detector The quantum mechanical explanation is that each particle passes through both slits and interferes with itself The Quantum Explanation The wavefunction of each particle is a probability wave which produces a probability interference pattern when it passes through the two slits.

23. Postulate 2: The motion of a nonrelativistic particle is governed by the Schrödinger equation Time-dependent S.E.: Time-independent S.E.: Time-independent S.E.: 1. It is a wave equation whose solutions displaysuperpositionandinterference effects. 2. It implies that time evolution is reversible. 3. It is verydifficult to solve for large systems (i.e. more than three particles).

24. Postulate 2: A quantum mechanical particle can tunnel through barriers rather than going over them. Classical ball Quantum ball Classical ball does not have enough energy to climb hill. Quantum ball tunnels through hill despite insufficient energy. This effect is the basis for the scanning tunneling electron microscope (STEM)

25. Postulate 2: Quantum particles take all paths. Classical picture Quantum picture The Schrodinger equation indicates that there is a nonzero probability for a particle to take any path Classical particles take a single path specified by Newton’s equations.

26. Postulate 3: Measurement of a quantum mechanical system is associated with some linear, Hermitian operator Ô. 1. It implies that certain properties can only achieve a discrete set of measured values 2. It implies that measurement is inherently probabilistic. 3. It implies that measurement necessarily alters the observed system.

27. Postulate 3: Even if the exact wavefunction is known, the outcome of measurement is inherently probabilistic Classical ball: Quantum ball: Before measurement + or After measurement For a known state, outcome is deterministic. For a known state, outcome is probabilistic.

28. Postulate 3: Measurement necessarily alters the observed system Classical Elephant: Quantum Elephant: Before measurement + After measurement State of the system is unchanged by measurement. Measurement changes the state of the system.

29. Postulate 3: Properties are actions to be performed, not labels to be read Classical Elephant: Quantum Elephant: Position = here Color = grey Size = large Position: The ‘position’ of an object exists independently of measurement and is simply ‘read’ by the observer ‘Position’ is an action performed on an object which produces some particular result In other words, properties like position or momentum do not exist independent of measurement!

30. Wave Properties of Particles • For photons: • De Broglie hypothesized that particles of well defined momentum also have a wavelength, as given above, the de Broglie wavelength

32. De Broglie’s Hypothesis • ALLmaterial particles possess wave-like properties, characterized by the wavelengthλB,related to the momentum p of the particlein the same way as for light Planck’s Constant Momentum of the particle de Broglie wavelength of the particle Frequency:

33. De Broglie’s Hypothesis • De Broglie’s waves are not EM waves • He called them “material” waves • λBdepends on the momentum and not on physical size of the particle • For a non-relativistic free particle: • Momentum is p = mv, here v is the speed of the particle • For free particle total energyE, is kinetic energy

34. “Construction” Particles From Waves • Particles are localized in space • Waves are extended in space. • It is possible to build “localized” entities from a superposition of number of waves with different values of k-vector. For a continuum of waves, the superposition is an integral over a continuum of waves with different k-vectors. • The wave then has a non-zero amplitude only within a limited region of space • Such wave is called “wave packet”

35. Wave Picture of Particle • Consider a wave packet made up of waves with a distribution of wave vectors k, A(k), at time t. • The spatial distribution at a time t given by:

36. Probability of the Particle • The probability of observing the particle between x and x + dx in each state is • Note that E0 = 0 is not a possible energy level. • The concept of energy levels, as first discussed in the Bohr model, has surfaced in a natural way by using waves.

37. Particle in a Box • A particle of mass m is trapped in a one-dimensional box of width L. • The particle is treated as a wave. • The box puts boundary conditions on the wave. The wave function must be zero at the walls of the box and on the outside. • In order for the probability to vanish at the walls, we must have an integral number of half wavelengths in the box. • The energy of the particle is . • The possible wavelengths are quantized which yields the energy: • The possible energies of the particle are quantized.

38. Crystal band structure

39. + + Crystallattice We define lattice points; these are points with identical environments Crystal = lattice + basis

40. Crystallattice

41. Crystallattice

42. Crystallattice Choice of origin is arbitrary - lattice points need not be atoms - but unit cell size should always be the same

43. Crystallattice Choice of origin is arbitrary - lattice points need not be atoms - but unit cell size should always be the same

44. Crystallattice This is NOT a unit cell even though they are all the same - empty space is not allowed!

45. Crystallattice – choice of the unit cell no yes