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ECE 874: Physical Electronics. Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University [email protected] Lecture 10, 02 Oct 12. Answers I can find:. Working tools:. Connection: conservation of energy and working tool 2: the Schroedinger equation.

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Ece 874 physical electronics

ECE 874:Physical Electronics

Prof. Virginia Ayres

Electrical & Computer Engineering

Michigan State University

[email protected]


Lecture 10 02 oct 12
Lecture 10, 02 Oct 12

VM Ayres, ECE874, F12


Answers I can find:

VM Ayres, ECE874, F12


Working tools
Working tools:

VM Ayres, ECE874, F12


Connection conservation of energy and working tool 2 the schroedinger equation
Connection: conservation of energy and working tool 2: the Schroedinger equation

VM Ayres, ECE874, F12


VM Ayres, ECE874, F12 Schroedinger equation


VM Ayres, ECE874, F12 Schroedinger equation


VM Ayres, ECE874, F12 Schroedinger equation


VM Ayres, ECE874, F12 Schroedinger equation


Two unknowns y x and e in ev from one equation
Two unknowns Schroedinger equationy(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.

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


Or equivalent Aexp Schroedinger equationikx + Bexp-ikx form

Infinite potential well

VM Ayres, ECE874, F12


With B = 0: tunnelling out of a finite well Schroedinger equation

VM Ayres, ECE874, F12


Finite potential well
Finite Potential Well: Schroedinger equation

(eV)

Electron energy: E > U0

Electron energy: E < U0

(nm)

Regions:

-∞ to 0

0 to a

a to +∞

VM Ayres, ECE874, F12


Finite potential well1
Finite Potential Well: Schroedinger equation

(eV)

Electron energy: E < U0

(nm)

Region:

0 to a

VM Ayres, ECE874, F12


Finite potential well2
Finite Potential Well Schroedinger equation

VM Ayres, ECE874, F12


Finite potential well3
Finite Potential Well: Schroedinger equation

(eV)

Electron energy: E < U0

(nm)

Regions:

-∞ to 0

a to +∞

VM Ayres, ECE874, F12


Finite potential well4
Finite Potential Well Schroedinger equation

VM Ayres, ECE874, F12


Finite potential well5
Finite Potential Well Schroedinger equation

New: e- goes away at ±∞

New boundary matching condition

New: e- exists outside of well region

VM Ayres, ECE874, F12


Finite potential well6
Finite Potential Well Schroedinger equation

Gives a decreasing exponential e-a|x| in this region

VM Ayres, ECE874, F12


Finite potential well7
Finite Potential Well Schroedinger equation

VM Ayres, ECE874, F12


Finite potential well8
Finite Potential Well Schroedinger equation

y-(x) and y0(x) are done to within A0. If you need A0, use Working Tool 3: the existence theorem, in the easy region: 0 < x < a.

VM Ayres, ECE874, F12


Finite potential well9
Finite Potential Well Schroedinger equation

To find B+ in terms of A0 to complete y+(x) add 2.41b and 2.41d, and re-arrange to get B+:

(2.41a)

(2.41b)

(2.41c)

(2.41d)

VM Ayres, ECE874, F12


Finite potential well10
Finite Potential Well Schroedinger equation

y-(x), y0(x) and y+(x) are done to within A0. If you need A0, use Working Tool 3: the existence theorem, in the easy region: 0 < x < a.

Wave functions that represent e- are found.

Now find its total energy E in eV.

VM Ayres, ECE874, F12


Finite potential well11
Finite Potential Well Schroedinger equation

.42)

VM Ayres, ECE874, F12


Finite potential well12
Finite Potential Well Schroedinger equation

.42)

This is basically the solution for E in eV.

VM Ayres, ECE874, F12


Finite potential well13
Finite Potential Well Schroedinger equation

VM Ayres, ECE874, F12


Finite potential well14
Finite Potential Well Schroedinger equation

Red: LHS curve

Blue: RHS curve

Solve graphically:

LHS = tan(…E)

RHS = polynomial (…E)

Where they intersect is the value for E in eV

E in eV

VM Ayres, ECE874, F12


Finite potential well15
Finite Potential Well Schroedinger equation

Red: LHS curve

Blue: RHS curve

Solve graphically:

LHS = tan(…E)

RHS = polynomial (…E)

Where they intersect is the value for E in eV

Quantized E1, E2, E3,.. for the finite well too, since tan(…E) repeats itself in multiples of p/2

E in eV

VM Ayres, ECE874, F12


Finite potential well16

E Schroedinger equationn in eV

Finite Potential Well:

These are the energies En for the e- in the well, but the values are consistent with the physical situation that the well has a finite height U0 and that the e- can tunnel into the out of well regions on either side.

(eV)

Electron energy: E < U0

(nm)

Regions:

-∞ to 0

0 to a

a to +∞

VM Ayres, ECE874, F12


Finite potential well17
Finite Potential Well Schroedinger equation

Advantage is: you scale to well height U0 and width a.

Note that width a only affects the LHS: the number/spacing of tan curves.

VM Ayres, ECE874, F12


Finite potential well18
Finite Potential Well Schroedinger equation

Red: LHS curve

Blue: RHS curve

Solve graphically:

LHS = tan(…E/U0)

RHS = polynomial (…E/U0)

Where they intersect are the values for En/U0 in eV

VM Ayres, ECE874, F12


(a) Shallow well U Schroedinger equation0, single intersection for E1

(b) Deeper well U0, more intersections for E1, E2, E3,….

(c) Comparison of finite (solid) and infinite (dotted) well energy levels En shows that the infinite well solution progressively over-estimates the higher En

VM Ayres, ECE874, F12


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