Homework #2
ECE 3450 “Semiconductor Devices”
Dr. John D. Cressler
Due Date: Wednesday, September 23rd
Topical Coverage: Quantum Theory and Energy Bands
[1] Why doesn't a fast-ball thrown at 100 mph by John Smoltz exhibit wave-like properties, as quantum theory guarantees? (I said fast ball not curve ball!). Answer this question quantitatively by calculating its de Broglie wavelength. Take the mass of the ball to be half a kilogram. Based on your result, and what you now know about real crystals (lattice constant etc.), is it even conceivable that we could do an experiment to demonstrate that the fast ball in fact has wave-like properties? Why or why not?
[2] Imagine that you are going to use a piece of silicon crystal in an experiment to clearly demonstrate that an electron has wavelike properties. How fast should be electrons be moving to accomplish what you’re after (in cm/s)?
[3] Consider an electron with the following wave function.
ψ(x) = A e i(3x) ,0≤ x≤ 3 ; assume A is real
Find the momentum of the electron.
[4] Following the text in Pierret's Advanced Semiconductor Fundamentals in Section 2.3.2 (your article # 4), solve the quantum mechanical problem of "particle in a 1-D box". Make your solution complete. Sketch the potentials, follow through the math at a level deeper than the text (i.e., fill in the missing steps), and understand your result. Do the following.
Consider your particle to be an electron caught between the two faces of the silicon unit cell. (ie, "m" and "a" are explicitly specified numbers). Solve the Schrodinger for this problem, apply the boundary conditions, and find the resultant wave function.
[5] Calculate the electron energy in this "toy" system for the first three electron quantization levels (in eV).
[6] Calculate the quantum mechanical momentum of the electron trapped in this box. For full credit you must carefully explain the result.
[7] Using our equation for the electron effective mass, prove that it indeed has units of mass.
[8] A hypothetical energy band can be fitted approximately to the equation
E(k) = E0 { 1 – exp(-2a2k2)}
where a is the lattice constant of the crystal. Calculate:
i) the effective mass at k=0
ii) the value of k for the maximum electron velocity
iii) the effective mass at the edge of the Brillouin zone
[9] The respective energy band diagrams of two different hypothetical semiconductor samples are shown at 300K. For each sample (a and b) answer the following. Assume both samples are grounded at x=L. For full credit you must shown how you got your answers.
i) Is the sample in equilibrium? Why or why not?
ii) Sketch the electrostatic potential as a function of position.
iii) Sketch the electric field as a function of position.
[10] Consider the band structure of two hypothetical semiconductors:
Semi. A Semi. B
i) Which semiconductor has the largest lattice constant? You must defend how you got your answer.
ii) Which semiconductor would you expect to have the largest electron mobility? You must defend how you got your answer.
iii) Which semiconductor would you expect to have the largest hole mobility? You must defend how you got your answer.
iv) How many equivalent conduction band minima do you expect for each semiconductor. You must defend how you got your answer.
v) Which semiconductor is direct gap, and which is indirect gap? You must defend how you got your answer.