2. Almost There! Interference and Review for 3 rd Hour Exam

3. Review The probability of finding a particle in a particular region within a particular time interval is found by integrating the square of the wave function: P (x,t) =  |  (x,t)| 2 dx =  |  (x)| 2 dx |  (x)| 2 dx is called the “probability density; the area under a curve of probability density yields the probability the particle is in that region When a measurement is made, we say the wave function “collapses” to a point, and a particle is detected at some particular location

4. Particle in a box  (x) = B sin (n  x/a)  (x) |  (x)| 2 n=2 n=3  Only certain wavelengths  = 2a/n are allowed  Only certain momenta p = h/ = hn/2a are allowed  Only certain energies E = p 2 /2m = h 2 n 2 /8ma 2 are allowed - energy is QUANTIZED  Allowed energies depend on well width

5. “Real-World” Wells Solution has non-trivial form, but only certain states (integer n) are solutions Each state has one allowed energy, so energy is again quantized Energy depends on well width a (confinement width) |  (x)| 2 n=1 n=2 x

6. Quantum wells An electron is trapped since no empty energy states exist on either side of the well

7. Escaping quantum wells Classically, an electron could gain thermal energy and escape For a deep well, this is not very probable. Given by Boltzmann factor.

8. Escaping quantum wells Thanks to quantum mechanics, an electron has a non-zero probability of appearing outside of the well This happens much more often than thermal escape if the wells are close together.

9. Tunneling and Interference Can occur when total particle energy is less than barrier height. Particle can be scattered back even when its energy is greater than barrier height. What affects tunneling probability? T  e –2kL k = [8  2m(E pot – E)] ½ /h

10. A tunnel diode According to quantum physics, electrons could tunnel through to holes on the other side of the junction with comparable energy to the electron This happens fairly often Applying a bias moves the electrons out of the p-side so more can tunnel in

11. The tunneling transistor As the potential difference increases, the energy levels on the positive side are lowered toward the electron’s energy Once the energy state in the well equals the electron’s energy, the electron can go through, and the current increases.

12. The tunneling transistor The current through the transistor increases as each successive energy level reaches the electron’s energy, then decreases as the energy level sinks below the electron’s energy

13. Quantum Entanglement (Quantum Computing) Consider photons going through beam splitters NO way to predict whether photon will be reflected or transmitted! (Color of line is NOT related to actual color of laser; all beams have same wavelength!)

14. Randomness Revisited If particle/probabilistic theory correct, half the intensity always arrives in top detector, half in bottom BUT, can move mirror so no light in bottom! (Color of line is NOT related to actual color of laser; all beams have same wavelength!)

15. Interference effects Laser light taking different paths interferes, causing zero intensity at bottom detector EVEN IF INTENSITY SO LOW THAT ONE PHOTON TRAVELS THROUGH AT A TIME What happens if I detect path with bomb? No interference, even if bomb does not detonate!

16. Interpretation Wave theory does not explain why bomb detonates half the time Particle probability theory does not explain why changing position of mirrors affects detection Neither explains why presence of bomb destroys interference Quantum theory explains both! –Amplitudes, not probabilities add - interference –Measurement yields probability, not amplitude - bomb detonates half the time –Once path determined, wavefunction reflects only that possibility - presence of bomb destroys interference

17. Quantum Theory meets Bomb Four possible paths: RR and TT hit upper detector, TR and RT hit lower detector (R=reflected, T=transmitted) Classically, 4 equally-likely paths, so prob of each is 1/4, so prob at each detector is 1/4 + 1/4 = 1/2 Quantum mechanically, square of amplitudes must each be 1/4 (prob for particular path), but amplitudes can be imaginary or complex! –e.g.,

18. Adding amplitudes Lower detector: Upper detector:

19. What wave function would give 50% at each detector? Must have |a| = |b| = |c| = |d| = 1/4 Need |a + b| 2 = |c+d| 2 = 1/2

20. Pictorial Representation of 3D Integration Concept using Wafer Bonding, * Figure adapted from IBM Corporation and used with permission. J. Lu et al

21. Broad band interconnect technology ---high speed data transfer Replacing electrical connection by optics: Modulators/switches: electro-optic, optic-optic Optical waveguides Data compression (software) Modulatorsguide Chip stack switches fiber Or: wireless! light

22. Oriented & interconnected nanotube networks—Ajayan et al –Local modification and Junction formation –Termination (cutting of structures) Focused Ions

23. DNA and a little more Ivar Giaever Rensselaer Polytechnic Institute and Applied BioPhysics, Inc. Troy, NY 12180 and Oslo Universitetet Blindern, Oslo

26. Wide Bandgap Semiconductors What is a wide bandgap semiconductor? Larger energy gap allows higher power and temperature operation and the generation of more energetic (i.e. blue) photons The III-nitrides (AlN, GaN and InN), SiC have recently become feasible. Other materials (like diamond) are being investigated. What are they good for?

27. How does a semiconductor laser work?

28. Stimulated vs. Spontaneous Emission (Cont.) Derived in 1917 by Einstein. (Required for thermal equilibrium was it was recognized that photons were quantized.) However, a “real” understanding of this was not achieved until the 1950’s.

29. Biased junction n-type p-type depleted region (electric field) Negative bias photon out

30. MOSFET The potential difference between drain and source is continually applied When the gate potential difference is applied, current flows Source Drain n-type p-type n-type Gate (Metal-Oxide-Semiconductor, Field-Effect Transistor)

31. Einstein to the Rescue Einstein suggested that light was emitted or absorbed in particle-like quanta, called photons, of energy, E = hf crest trough If an electron absorbs one of these photons, it gets the entire hf of energy. If that energy is larger than the work function of the metal, the electron can leave; if not, it can’t: K max = E abs –  = hf - 

32. Bipolar Junction Transistor n-type p-typen-type Emitter Base Collector increasing electron energy increasing hole energy

33. Bipolar Junction Transistor http://hyperphysics.phy-astr.gsu.edu/hbase/solids/trans.html#c1

34. NOT Gate - the simplest case  Put an alternate path (output) before a switch.  If the switch is off, the current goes through the alternate path and is output.  If the switch is on, no current goes through the alternate path.  So the gate output is on if the switch is off and off if the switch is on. Output Dump Input Switch

35. AND - slightly more complicated  AND gate returns a signal only if both of its two inputs are on.  Use the NAND output as input for NOT  If both inputs are on, the NOT input is off, so the AND output is on.  Else the NOT input is on, so the output is off. Dump Input Switch Output