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

2. Lecture 16, 23 Oct 12
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3. Effective mass: How: practical discussion:
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4. Reminder: how you got the E-k curves: Kronig-Penney model allowed energy levels, Chp. 03:
LHS
RHS
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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5. k = 0
k = ± p
a + b
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6. (b)
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7. (b)
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8. Matlab can do numerical derivatives
Get: the E-k curves.
Matlab can do numerical derivatives
Note that the effective mass m* isn’t a single number.
Note also that a + b = aBl varies depending on what direction you move in, so there are more curves than are on this single ± direction chart.
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9. Which band has the sharpest curvature d2E/dk2?
Get: the E-k curves.
Region of biggest change of tangent = greatest curvature: the parabolas shown.
Example problem:
Which band has the sharpest curvature d2E/dk2?
Which band has the lightest effective mass?
Which band has the heaviest effective mass?
Where in k-space, for both?
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10. Which band has the sharpest curvature d2E/dk2? Band 4
Get: the E-k curves.
Region of biggest change of tangent = greatest curvature: the parabolas shown.
Example problem:
Which band has the sharpest curvature d2E/dk2? Band 4
Which band has the lightest effective mass?
Which band has the heaviest effective mass? Band 1: broadest = least curvature divide by smallest number = heaviest m*
Where in k-space, for both?
At k= 0 called the G point
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11. VM Ayres, ECE874, F12

12. Where in k-space, for both?
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13. Where in k-space, for both? m*A at G  k = 0
m*B at about ½ way between G and X in [100] direction:
k = 0
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14. k = p/aBl = p/aLC at end of Zone 1
a + b = aBl
aBl for [100] = aLC
k = p/aBl = p/aLC at end of Zone 1
This is X for [100]
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15. Where in k-space, for both? m*A at G  k = 0
m*B at about ½ way between G and X in [100] direction at k = p/2 aLC
k = 0
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16. Assume T = 300K and it doesn’t change
Ec = Egap = constant at a given T
Hint: compare the answers for b = 0 and b ≠ 0 in (a)
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17. Pick correct curve:
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18. Pick conduction or valence bands::
E – Ec (eV)
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19. Pick conduction minima. Where in k-space are they?
E – EV (eV)
<111>
L G X
<100>
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20. Pick conduction minima. Where in k-space are they?
G at k = 0
L at k = p/aBl for <111>
Could work out the aBl distance between atomic cores in a <111> direction if needed.
Not needed to finish answering the question.
E – EV (eV)
<111>
L G X
<100>
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21. Note that the effective mass m* isn’t a single number.
Go back to here:
Note that the effective mass m* isn’t a single number.
Note also that a + b = aBl varies depending on what direction you move in, so there are more curves than are on this single ± direction chart.
VM Ayres, ECE874, F12

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26. (From practical to fundamental!)
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27. In 3 D:
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28. Write this in 2D: all three parts. Integrate a -> v -> r.
Vector r (t) is the direction. The final answer contains time t.
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29. Then a = dv/dt for dvx/dt and dvy/dt
Start with [m*ij]
Then F = qE
Then a = dv/dt for dvx/dt and dvy/dt
Integrate with respect to time, 2x’s, to get x(t) and y(t).
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30. k = 0
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31. 1D: Any one of these parabolas could be modelled as:
Region of biggest change of tangent = greatest curvature: the parabolas shown.
1D: Any one of these parabolas could be modelled as:
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32. For any of these parabolas:
Region of biggest change of tangent = greatest curvature: the parabolas shown.
3D: <111> + <100>
E – EV (eV)
<111>
L G X
<100>
For any of these parabolas:
There’s a major axis but also two minor ones
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33. E – EV (eV) <111> L G X <100> Same: truncate 1/2
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34. k = 0
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