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

VM Ayres, ECE874, F12 Lecture 08, 25 Sep 12

VM Ayres, ECE874, F12 Example problem: what is the average value of the x-component of linear momentum p for a nearly free electron in GaAs, described by the traveling wave: Travelling wave moving R

VM Ayres, ECE874, F12

No use made of GaAs: purely wave-like.

VM Ayres, ECE874, F12 No use made of GaAs: purely wave-like.

VM Ayres, ECE874, F12 Travelling wave moving R

VM Ayres, ECE874, F12

Example problem: for this nearly free electron in GaAs, write a simple statement of conservation of energy (no calculation required): Travelling wave moving R

VM Ayres, ECE874, F12

= h bar k Dispersion diagram: E-k:

VM Ayres, ECE874, F12 This electron: Satisfies conservation of energy: For physical situation U(x,y,z) = 0.

VM Ayres, ECE874, F12 Chp. 02: 3 important examples of electron showing wavelike properties 1. Free electron An electron between scattering events during transport in a semiconductor is often nearly free Motivation: low heat transistors (hard in 3D due to plenty of scattering) 2. Electron in an infinite potential well 3. Electron in a finite potential well 4. Pr. 2.7: Electrons in a triangular well = realistic Motivation: these are all about Quantum well lasers: Arai article: Transitions between quantized energy levels  E = hc/  what you see is light of precise wavelength 

VM Ayres, ECE874, F12 2. Electron in an infinite potential well Describing its wave properties:  (x,y,x,t) Several choices: physical situation in conservation of energy selects the right one

VM Ayres, ECE874, F12 2. Electron in an infinite potential well U(x,y,z) => U(x) a

VM Ayres, ECE874, F12 2. Electron in an infinite potential well U(x,y,z) => U(x) a

VM Ayres, ECE874, F12 Expectations:

VM Ayres, ECE874, F12 Expectations:

VM Ayres, ECE874, F12 Use conservation of energy to find 2 things: 1.  (x): correct wave description of electron 2. total energy E For: U(x) eV a nm0

VM Ayres, ECE874, F12 Worked through pp on board:

VM Ayres, ECE874, F12 Worked through pp on board:

VM Ayres, ECE874, F12 Useful consequence: clean laser light emission from a quantum well. Example: GaAs: 1.43 eV In a transition from the first conduction band energy level to the first valence (bonding) band energy level, en electron will loose this amount of energy  E (next page).

VM Ayres, ECE874, F12 M. Arai, et al 19 th IPRM Conference, Matsue, Japan

VM Ayres, ECE874, F12 Also have discreet energies AND momentum value on the dispersion diagram: 1.43 eV Conduction band Valence band