ECE 874: Physical Electronics

ECE 874:Physical Electronics

Prof. Virginia AyresElectrical & Computer EngineeringMichigan State [email protected]

VM Ayres, ECE874, F12

Lecture 10, 02 Oct 12


Answers I can find:


Working tools:


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






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.

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.


Or equivalent Aexpikx + Bexp-ikx form

Infinite potential well


With B = 0: tunnelling out of a finite well


Finite Potential Well:

-∞ to 0 a to +∞0 to a

(nm)

(eV)

Electron energy: E > U0

Electron energy: E < U0

Regions:



0 to a

(nm)

(eV)


Region:


Finite Potential Well



-∞ to 0 a to +∞

(nm)

(eV)


Regions:





New: e- exists outside of well region

New: e- goes away at ±∞

New boundary matching condition



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



0

0kAB



0

0kABA

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



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

(2.41a)

(2.41b)

(2.41c)

(2.41d)



-(x), 0(x) and +(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.



.42)



.42)

This is basically the solution for E in eV.





Solve graphically:

LHS = tan(…E)

RHS = polynomial (…E)

Where they intersect is the value for E in eV

E in eV

Red: LHS curveBlue: 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

E in eV




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.

-∞ to 0 a to +∞0 to a

(nm)

(eV)


Regions:

En in eV



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.



Solve graphically:

LHS = tan(…E/U0)

RHS = polynomial (…E/U0)

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



(a) Shallow well U0, 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

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ECE 874: Physical Electronics