ECE 874:Physical Electronics

Prof. Virginia AyresElectrical & Computer EngineeringMichigan State Universityayresv@msu.edu

VM Ayres, ECE874, F12

Lecture 11, 04 Oct 12

VM Ayres, ECE874, F12

Answers I can find:

VM Ayres, ECE874, F12

Working tools:

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Two unknowns (x) and E in eV from one equation:

1. You can find (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 (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

Or equivalent Aexpikx + Bexp-ikx form

Infinite potential well

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With B = 0: tunnelling out of a finite well

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Finite Potential Well:

-∞ to 0 a to +∞0 to a

(nm)

(eV)

Electron energy: E > U0

Electron energy: E < U0

Regions:

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Infinite Potential Well:

-∞ to 0 a to +∞0 to a

(nm)

U (eV) = +∞

Electron energy: E < U0

Regions:

U (eV) = +∞

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Free (between scattering events) particle (e- in I):

-∞ to +∞Region:

Electron energy: E > U0

U (eV) = 0

(nm)

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For all three situations, found:

- (x)

- E (free) or En (infinite and finite wells)

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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 bandgapB) U0 = 1.4 eV = just under the size of the bandgapC) What is the practical meaning of the limit: = E/U0, 0 < < 1?

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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..

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

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Expected wavefunctions in each of three regions are easy:

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Energy levels: set up the graphical solution:

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units

VM Ayres, ECE874, F12

VM Ayres, ECE874, F12

VM Ayres, ECE874, F12

Example problem: Find (x) for a mixed U0 situation modelled as an infinite/triangular well

VM Ayres, ECE874, F12

VM Ayres, ECE874, F12