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

2. Lecture 14, 16 Oct 12
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3. - “unit cell” of a periodic potential, p.52 - well width I
Three different “a”:
- lattice constant
- “unit cell” of a periodic potential, p.52
- well width I
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4. Lattice constant a of the Unit cell GaAs: 5.65 Ang
“unit cell” a of a 1-D periodic potential
Block theorem
Well width a
Kronig-Penney model for a 1-D periodic potential
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5. Example problem: An electron is moving along the [110] direction in GaAs, lattice constant = 5.65 Ang.
(a) Write down both versions of the Block theorem explicitly solving for the “unit cell” of the periodic potential in terms of the lattice constant.
(b) Draw a model of the transport environment using the Kronig-Penney model where the well width is 20% of the “unit cell” of a periodic potential. Write the dimensions in terms of the lattice constant. +z +x +y
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6. Face diagonal distance = ✔2 a
Example problem: An electron is moving along the [110] direction in GaAs, lattice constant = 5.65 Ang.
[110]
Face diagonal distance = ✔2 a
Distance between atoms = ✔2 a/2 +z +x +y
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7. VM Ayres, ECE874, F12

8. (b) +z +z +x [110] +x [110] +y +y Rotate [110] to go “straight”
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9. (b) b + a = aBl = ✔2 aLC/2 = 3.995 Ang b = 0.8 (aBl = 3.995 Ang)
aKP = 0.2 (aBl = Ang)
= Ang b
aKP
[110]
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10. Finite Well boundary conditions, Chp. 02:
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11. Finite Well allowed energy levels, Chp. 02:
Graphical solution for number and values of energy levels E1, E2,…in eV.
a is the finite well width.
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12. Similar for Kronig-Penney model but new periodicity requirements:
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13. Kronig-Penney model allowed energy levels, Chp. 03:
Graphical solution for number and values of energy levels E1, E2,…in eV.
a = width of well, b = width of barrier, a + b = Block periodicity aBl
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14. Kronig-Penney model allowed energy levels, Chp. 03:
Graphical solution for number and values of energy levels E1, E2,…in eV.
Also have values for k from RHS.
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15. Example problem:
(a) What are the allowed (normalized) energies and also the forbidden energy gaps for the 1st-3rd energy bands of the crystal system shown below?
(b) What are the corresponding (energy, momentum) values? Take three equally spaced k values from each energy band.
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16. k = 0
0.5
k = ± p
a + b
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17. (a)
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18. (b)
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19. “Reduced zone” representation of allowed E-k states in a 1-D crystal
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20. k = 0
k = ± p
a + b
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