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

Lecture 11, 04 Oct 12 VM Ayres, ECE874, F12

Answers I can find: VM Ayres, ECE874, F12

Working tools: VM Ayres, ECE874, F12

Two unknowns y(x) and E in eV from one equation: 1. You can find y(x) by inspection whenever the Schroedinger equation takes a form with a known solution like and exponential. The standard form equation will also give you one relationship for kx that contains E in eV. 2. Matching y(x) at a boundary puts a different condition on kx and setting kx = kx enables you to also solve for E in eV. VM Ayres, ECE874, F12

Infinite potential well Or equivalent Aexpikx + Bexp-ikx form Infinite potential well VM Ayres, ECE874, F12

With B = 0: tunnelling out of a finite well VM Ayres, ECE874, F12

Finite Potential Well: (eV) Electron energy: E > U0 Electron energy: E < U0 (nm) Regions: -∞ to 0 0 to a a to +∞ VM Ayres, ECE874, F12

Infinite Potential Well: U (eV) = +∞ U (eV) = +∞ Electron energy: E < U0 (nm) Regions: -∞ to 0 0 to a a to +∞ VM Ayres, ECE874, F12

Free (between scattering events) particle (e- in I): Electron energy: E > U0 U (eV) = 0 (nm) Region: -∞ to +∞ VM Ayres, ECE874, F12

For all three situations, found: -. y(x) - For all three situations, found: - y(x) - E (free) or En (infinite and finite wells) VM Ayres, ECE874, F12

Example problem: Find energy levels in a finite model for a SQW: Consider a SQW of width a = 10 nm that is fabricated in GaAs that operates at 300K. The SQW is modelled as a finite well. How many energy levels for an e- exist for: A) U0 = 0.7 eV = half the size of the bandgap B) U0 = 1.4 eV = just under the size of the bandgap C) What is the practical meaning of the limit: x = E/U0, 0 < x < 1? VM Ayres, ECE874, F12

Finite Potential Well Advantage is: you scale to important parameters: the height U0 and width a. Note: Width a only affects the LHS: the number/spacing of tan curves. Height U0 affects both sides but practical advantage on RHS plot.. VM Ayres, ECE874, F12

Example problem: Find y(x) for a mixed U0 situation modelled as an infinite/finite well. Consider the case where E < U0-RHS. VM Ayres, ECE874, F12

Expected wavefunctions in each of three regions are easy: VM Ayres, ECE874, F12

Energy levels: set up the graphical solution: VM Ayres, ECE874, F12

units VM Ayres, ECE874, F12

VM Ayres, ECE874, F12

VM Ayres, ECE874, F12

Example problem: Find y(x) for a mixed U0 situation modelled as an infinite/triangular well VM Ayres, ECE874, F12

VM Ayres, ECE874, F12

VM Ayres, ECE874, F12