ECE 4339: Physical Principles of Solid State Devices

1. ECE 4339: Physical Principles of Solid State Devices 1A: Intro to Quantum Mechanics Len Trombetta Summer 2007 ECE 4339 L. Trombetta

2. Background One of the foundations for understanding the electrical and optical properties of semiconductors is Quantum Mechanics (QM). At the level of ECE 4339 we will not need much QM, but we will require familiarity with some basic ideas. Goal for this Lecture To figure out what Quantum Mechanics is all about. ECE 4339 L. Trombetta

3. The Electromagnetic Spectrum This information will be useful for several aspects of the course… http://kingfish.coastal.edu/marine/Animations/Images/Electromagnetic-Spectrum-2.png ECE 4339 L. Trombetta

4. Planck’s Hypothesis Blackbody radiation: An object gives off radiation over a range of frequencies that is temperature-dependent. In the early 1900’s, physicists studied this phenomenon using idealized structures called “blackbodies”, which were cavities designed to absorb (almost) all of the radiation incident on them. Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles; (2ed) Wiley and Sons, 1985 ECE 4339 L. Trombetta

5. Problem: The classical predictions for the energy density rT (energy per unit volume per Hz) of the radiation were way off (dashed line). Planck’s Solution: The energy of the radiation emitted by the blackbody is not continuous, that is, it cannot take arbitrary values (as one thinks of it in classical EM). Instead, it is “quantized”, i.e., it can be emitted only as discrete values as follows: E = hn, 2hn, 3hn, … ECE 4339 L. Trombetta

6. DE = hn The difference in successive values is thus h = 6/63 x 10-34 J-s is Planck’s constant n is the frequency of the radiation ECE 4339 L. Trombetta

7. Einstein’s Contribution Einstein said that the idea of radiation coming in discrete values is not limited to radiation emitted by a blackbody, but is true of electromagnetic radiation in general. He argued that the energy of radiation was emitted in bundles and traveled through space in packets called “photons”. Important Idea: We think of EM radiation as traveling in waves, but in some circumstances, it behaves like a particle (a “photon”). So is everything Dr. Williams told us WRONG? Hang on… ECE 4339 L. Trombetta

8. Is there an experiment that proves this? Funny you should ask: Is there an experiment that proves this? Funny you should ask: The Photoelectric Effect Light shining through the window strikes the metal target “A”, causing emission of electrons. We can measure the intensity of the electrons that reach “B” (with the Galvanometer G) as a function of their energy (by adjusting the potential V). Eisberg and Resnick, Fig. 2.1 The results… ECE 4339 L. Trombetta

9. Observations: • The potential needed to stop the most energetic electrons is Vo. This value does not depend on the intensity i of the light. • The saturation current depends proportionately on the intensity i. Classical EM predicts that if we double the intensity, we will give the electron double the energy, so Vo should double. That doesn’t happen. ECE 4339 L. Trombetta

10. Somebody in the audience: calculate the slope from the data. Is it what you expected? Observations: • For frequencies below no, we get no electrons emitted from the metal, no matter what the voltage is. • The stopping potential, and therefore the energy, is directly proportional to frequency, just as Planck predicted. Classical EM predicts that no matter what the intensity, if we shine light long enough, we will give the electron enough energy to leave the metal. That doesn’t happen. ECE 4339 L. Trombetta

11. Einstein’s Interpretation of the Photoelectric Effect Einstein said that the incident light was behaving like particles (i.e., photons). They had energy E = hn, just as Planck had said. That explains several things: • Doubling the intensity of the light doubles the current (the number of photons striking the metal) but doesn’t change the energy of the photons, so the electron energy (and Vo) doesn’t change. • There is a minimum energy needed to eject electrons. If n is too small, so is E, and no electrons are emitted. Einstein won the Noble Prize for Physics in 1921 for this. Click here… ECE 4339 L. Trombetta

12. Interpretation of the Photoelectric Effect F is the minimum energy necessary to get an electron out of the metal. This is the metal workfunction. The vertical axis represents electron energy. energy E = 0If there were an electron here, it would not be bound, and it would not be moving. F This electron is not very tightly bound. These are the energies that are available for electrons; electrons here are bound to the metal and can be viewed as having negative total energy. This electron is very tightly bound. ECE 4339 L. Trombetta

13. Maximum electron energy left after being ejected from the metal is Emax = hn – F = qVo This electron has been given energy hn by the photon; it is more than enough to “escape” from the metal, so the electron has energy Eel left over. This electron came from the least tightly bound level. It has the maximum energy left over: eVo. qVo energy Eel E = 0 Suppose incident photons have this much energy: hn incident photons F is the minimum escape energy (workfunction). F hn This electron would need more energy to get out than the photon can give it, so it will stay in the metal. If the electron is bound by an energy less than hn - F, it will have energy left over when it gets out. So for example it will be moving, and have kinetic energy Eel = mv2/2. ECE 4339 L. Trombetta

14. Another experiment confirming this idea is the Compton effect: Data from this experiment looked strange… …until Compton provided this explanation. It looks like photons are real !! Things are not looking good for Dr. Williams ! But hang in there… ECE 4339 L. Trombetta

15. An aside: Energy units. 1 eV = 1.6 x 10-19 J 1 Joule (J) is 1 kg-m2/s2. 1 eV (electron volt) is the energy gained by an electron after acceleration through a potential difference of 1 V. This is a natural unit for energy in solid state devices, where currents are dependent on electron energies. Now the slope of the photoelectric stopping potential vs. frequency should make sense… ECE 4339 L. Trombetta

16. At about the same time, the Bohr model of the atom was being developed. It was motivated by observation of atomic spectra… “The Balmer lines are designated by H with a Greek subscript in order of decreasing wavelength. Thus the longest wavelength Balmer transition is designated H with a subscript alpha, the second longest H with a subscript beta, and so on.” ECE 4339 L. Trombetta http://csep10.phys.utk.edu/astr162/lect/light/absorption.html

17. The de Broglie Hypothesis De Broglie’s idea : Waves sometimes behave like particles, and particles sometimes behave like waves. Particles with momentum p (= mv in classical physics) have a wavelength where l is the particle’s wavelength. A quote from de Broglie: “Two seemingly incompatible conceptions can each represent an aspect of the truth ... They may serve in turn to represent the facts without ever entering into direct conflict.” In Dialectica, taken from http://www.gap-system.org/~history/Quotations/Broglie.html ECE 4339 L. Trombetta

18. Fundamental Idea of Quantum Mechanics Everything is simultaneously a particle and a wave. ECE 4339 L. Trombetta

19. Consequences of the Fundamental Postulate • Heisenberg Uncertainty Relation • Probabilistic Nature of Physical Measurements ECE 4339 L. Trombetta

20. Q: How do we resolve the notion of particles and waves being the same thing? A: Fourier synthesis… Ref: Eisberg and Resnick So Dr. Williams didn’t have it “wrong” after all. He just didn’t know the whole story !! ECE 4339 L. Trombetta

21. The Schrödinger Wave Equation - time-dependent SWE We can say with in which case we can simplify to - time-independent SWE • Idea: • Solution of SWE provides Y. • In “bounded” systems (where the particle cannot escape), solutions exist only for certain values of the total energy E. ECE 4339 L. Trombetta

22. Sample problem: Particle in a box ECE 4339 L. Trombetta

23. Sample problem: Tunneling ECE 4339 L. Trombetta

24. What do I need to know from this presentation? • You should be familiar with… • …the figures explaining the photoelectric effect results; • …the de Broglie formula for the wavelength of a particle; • …the conversion of energy units to eV; • …the main results of our simple SWE examples: • energy is quantized when particles are “bound”; • because of the wavelike nature of particles, they can “tunnel” through barriers. ECE 4339 L. Trombetta