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

VM Ayres, ECE874, F12 Lecture 15, 18 Oct 12

VM Ayres, ECE874, F12 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 = ±  a + b

VM Ayres, ECE874, F k = 0 k = ±  a + b

VM Ayres, ECE874, F12 (a)

VM Ayres, ECE874, F12 (b)

VM Ayres, ECE874, F12 “Reduced zone” representation of allowed E-k states in a 1-D crystal

VM Ayres, ECE874, F12 k = 0 k = ±  a + b

VM Ayres, ECE874, F12 (b)

VM Ayres, ECE874, F12 “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, ECE874, F12 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, ECE874, F12 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, ECE874, F12 The missing band: Band 2

VM Ayres, ECE874, F12

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, ECE874, F12 Note that at 0 and ±  /(a+b) the tangent to each curve is flat: dE/dk = 0

VM Ayres, ECE874, F12 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, ECE874, F12

(b)

VM Ayres, ECE874, F12

 What happens to the e- in response to the application of an external force: example: a Coulomb force F = qE (Pr. 3.5):

VM Ayres, ECE874, F12 (d)

VM Ayres, ECE874, F12 (d) [100] type 6 of these type 8 of these Warning: you will see a lot of literature in which people get careless about versus [specific direction] SymmetricConduction energy bands

VM Ayres, ECE874, F12 (d) and type transport directions certainly have different values for a Block spacings of atomic cores. The , X, and L labels are a generic way to deal with this.

VM Ayres, ECE874, F12 Two points before moving on to effective mass: Kronig-Penney boundary conditions Crystal momentum, the Uncertainty Principle and wavepackets

VM Ayres, ECE874, F12 Boundary conditions for Kronig-Penney model: Can you write these blurry boundary conditions without looking them up?

VM Ayres, ECE874, F12 Locate the boundaries: [transport direction p 56] ba KP a KP + b = a Block -ba0 a

VM Ayres, ECE874, F12 Locate the boundaries: into and out of the well. [transport direction p 56] ba KP a KP + b = a Block -ba0 a

VM Ayres, ECE874, F12 Boundary conditions for Kronig-Penney model, p. 57: Is the a in these equations a KP or a Bl ?

VM Ayres, ECE874, F12 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, ECE874, F12 Two points before moving on to effective mass: Kronig-Penney boundary conditions Crystal momentum, the Uncertainty Principle and wavepackets

VM Ayres, ECE874, F12

Chp. 04: learn how to find the probability that an e- actually makes it into - “occupies” - a given energy level E.

VM Ayres, ECE874, F12 k2k2 k  wavenumber Chp. 02

VM Ayres, ECE874, F12 Suppose U(x) is a Kronig-Penney model for a crystal.

VM Ayres, ECE874, F12 On E-axis: Allowed energy levels in a crystal, which an e- may occupy h bar k = crystal momentum 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.