ECE 874 Physical Electronics Prof Virginia Ayres Electrical

ECE 874: Physical Electronics Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University ayresv@msu. edu

Lecture 15, 18 Oct 12 VM Ayres, ECE 874, F 12

Example problem: (a) What are the allowed (normalized) energies and also the forbidden energy gaps for the 1 st-3 rd energy bands of the crystal system shown below? (b) What are the corresponding (energy, momentum) values? Take three equally spaced k values from each energy band. k=0 k=± p a+b VM Ayres, ECE 874, F 12

0. 5 k=0 k=± p a+b VM Ayres, ECE 874, F 12

(a) VM Ayres, ECE 874, F 12

(b) VM Ayres, ECE 874, F 12

“Reduced zone” representation of allowed E-k states in a 1 -D crystal VM Ayres, ECE 874, F 12

k=0 k=± p a+b VM Ayres, ECE 874, F 12

(b) VM Ayres, ECE 874, F 12

“Reduced zone” representation of allowed E-k states in a 1 -D crystal This gave you the same allowed energies paired with the same momentum values, in the opposite momentum vector direction. Always remember that momentum is a vector with magnitude and direction. You can easily have the same magnitude and a different direction. Energy is a scalar: single value. VM Ayres, ECE 874, F 12

Can also show the same information as an “Extended zone representation” to compare the crystal results with the free carrier results. Assign a “next” k range when you move to a higher energy band. VM Ayres, ECE 874, F 12

Example problem: There’s a band missing in this picture. Identify it and fill it in in the reduced zone representation and show with arrows where it goes in the extended zone representation. VM Ayres, ECE 874, F 12

The missing band: Band 2 VM Ayres, ECE 874, F 12

VM Ayres, ECE 874, F 12

Notice that upper energy levels are getting closer to the free energy values. Makes sense: the more energy an electron “has” the less it even notices the well and barrier regions of the periodic potential as it transports past them. VM Ayres, ECE 874, F 12

Note that at 0 and ±p/(a+b) the tangent to each curve is flat: d. E/dk = 0 VM Ayres, ECE 874, F 12

A Brillouin zone is basically the allowed momentum range associated with each allowed energy band Allowed energy levels: if these are closely spaced energy levels they are called “energy bands” Allowed k values are the Brillouin zones Both (E, k) are created by the crystal situation U(x). The allowed energy levels are occupied – or not – by electrons VM Ayres, ECE 874, F 12

VM Ayres, ECE 874, F 12

(b) VM Ayres, ECE 874, F 12

VM Ayres, ECE 874, F 12

What happens to the e- in response to the application of an external force: example: a Coulomb force F = q. E (Pr. 3. 5): VM Ayres, ECE 874, F 12

(d) VM Ayres, ECE 874, F 12

(d) Conduction energy bands <111> type 8 of these <100> type 6 of these Symmetric [100] Warning: you will see a lot of literature in which people get careless about <direction type> versus [specific direction] VM Ayres, ECE 874, F 12

(d) <111> and <100> type transport directions certainly have different values for a. Block spacings of atomic cores. The G, X, and L labels are a generic way to deal with this. VM Ayres, ECE 874, F 12

Two points before moving on to effective mass: l l Kronig-Penney boundary conditions Crystal momentum, the Uncertainty Principle and wavepackets VM Ayres, ECE 874, F 12

Boundary conditions for Kronig-Penney model: Can you write these blurry boundary conditions without looking them up? VM Ayres, ECE 874, F 12

Locate the boundaries: a. KP + b = a. Block b a. KP [transport direction p 56] -b a 0 a -b VM Ayres, ECE 874, F 12

Locate the boundaries: into and out of the well. a. KP + b = a. Block b a. KP [transport direction p 56] -b a 0 a -b VM Ayres, ECE 874, F 12

Boundary conditions for Kronig-Penney model, p. 57: Is the a in these equations a. KP or a. Bl? VM Ayres, ECE 874, F 12

Boundary conditions for Kronig-Penney model, p. 57: Is the a in these equations a. KP or a. Bl? It is a. KP. VM Ayres, ECE 874, F 12

Two points before moving on to effective mass: l l Kronig-Penney boundary conditions Crystal momentum, the Uncertainty Principle and wavepackets VM Ayres, ECE 874, F 12

VM Ayres, ECE 874, F 12

Chp. 04: learn how to find the probability that an e- actually makes it into “occupies” - a given energy level E. VM Ayres, ECE 874, F 12

k 2 k wavenumber Chp. 02 VM Ayres, ECE 874, F 12

Suppose U(x) is a Kronig-Penney model for a crystal. VM Ayres, ECE 874, F 12

On E-axis: Allowed energy levels in a crystal, which an e- may occupy So a dispersion diagram is all about crystal stuff but there is an easy to understand connection between crystal energy levels E and e- ‘s occupying them. The confusion with momentum is that an e-’s real momentum is a particle not a wave property. Which brings us to the need for wavepackets. http: //en. wikipedia. org/wiki/Crystal_momentum hbark = crystal momentum VM Ayres, ECE 874, F 12