Physics 322a-2007 Walter F. Smith Practice Questions for Final Exam

You should be able to do each of these problems (including all sub-parts) in about 15 minutes or less. If it takes you longer, you have not mastered the associated material thoroughly enough. Note: Of course, these problems don’t cover exactly the same topics as the questions on the real exam. Therefore, doing well on these practice problems does not insure that you are adequately prepared for the exam. However, doing poorly on these problems does mean that you need to study more. You will need a ruler marked in mm and cm.

Note: These practice questions mostly emphasize the material since the last exam. However, the actual final exam will include questions from the entire course. So, you should review exams 1 and 2 and the associated practice exams.

Some useful data:

k = 8.99₉ Nm²/C² e_o = 8.85_-12 C²/Nm² = 1/(4pk) Charge of electron = -e, where e = 1.60_-19 C

c = 3.00₈ m/s m_o = 4p_-7 N/A² Mass of electron = m_e = 9.11_-31 kg

Mass of proton = m_p = 1.67_-27 kg

Silicon:

e_g = 1.12 eV, k = 11.7, m_e (room temp) = 0.135 m²/V-s, m_h (room temp) = 0.048 m²/V-s

1. (7 points) Describe what happens to the band structure diagram when two clean metal surfaces with different work functions are brought together to form a junction. Your discussion should include a microscopic consideration of the charge distribution very close to the surface, and how this alters the bands very close to the surface. Also, explain why no diode behavior is seen in the I-V curve of such a junction.

2. (12 points) We did not have time to discuss it, but pn junctions can also be used as temperature sensors, and are in fact used this way, especially for scientific applications at low temperatures. Someone hands you a pn junction diode which has been designed for this purpose, but tells you nothing else about it. Explain quantitatively how you would convert measured voltage(s) and/or current(s) at some unknown temperature (you must specify what measurements are to be taken) into a value for the temperature. You may assume that you have available good voltmeters and current meters, and good voltage supplies and current supplies. (The latter supply a constant, pre-set current, no matter how much voltage is required to do so.) Hint: One solution involves making measurements first under reverse bias and then under forward bias.

3. (10 points) The resistivity of intrinsic silicon at room temperature is 3.4₅ W cm at room temperature. The resistivity for a particular sample (call it “sample A”) of n-doped silicon is measured to be r = 1.2 W cm at room temperature. A second sample (“sample B”) is also n-doped, and has r = 0.35 W cm at room temperature. What is the ratio of the density of holes in sample A to that in sample B? Show your work.

4. (20 points) Experimentally, one observes that the effective mass of electrons in most materials is temperature-dependent. Use the fact that thermal expansion causes the atoms to move further apart as the material is heated to explain this using the tight-binding model. As part of your explanation, predict whether m* should increase or decrease as T is increased. You may make any simplifying assumptions you like (e.g. feel free to treat a 1D case). Hint: Recall that , where j₁ and j₂ are atomic wavefunctions of neighboring atoms.

5. For aluminum (which is trivalent fcc), show that the free-electron Fermi sphere completely encloses (but just barely) the 1BZ.

Some facts that may or may not be useful:

The 1BZ for fcc is shown there.

The distance from the origin to the X point is .

The distance from the origin to the L point is .

The distance from the origin to the W point is .