1. ECE 802-604: Nanoelectronics
Prof. Virginia Ayres
Electrical & Computer Engineering
Michigan State University

2. Lecture 12, 08 Oct 13 In Chapter 02 in Datta:
Transport: current I = GV
V = IR => I = GV
Velocity
Energy levels M  M(E)
Conductance G = GC in a 1-DEG
Example Pr. 2.1: 2-DEG-1-DEG-2-DEG
Example: 3-DEG-1-DEG-3-DEG
Transmission probability: the new ‘resistance’
How to evaluate the Transmission/Reflection probability
T(N) for multiple scatterers
T(L) in terms of a “how far do you get length” L0
How to correctly measure I = GV
Landauer-Buttiker: all things equal
3-, 4-point probe experiments: set-up and read out
Example: 3-point probe (a) and (b)
Example: 4-point probe (a) and (b)
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3. Lecture 12, 08 Oct 13 In Chapters 02 and 03 in Datta:
How to correctly measure I = GV
Add scattering to Landauer-Buttiker
In Dresselhaus:
Carbon bond hybridizations
Graphene
Carbon nanotubes
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4. Lec11: Example: 3 point probe-a
Question:
How would you find R for this configuration?
Answer:
a/ Choose: V3 = 0
b/ Choose: ideal I2 = 0
c/ From I-V 2x2 matrix equations, identify the R that you want as: =
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5. Lec 11:Example: 3 point probe-b
Question:
How would you find R for this configuration?
Answer:
a/ Choose: V3 = 0
b/ Choose: ideal I1 = 0
c/ From I-V 2x2 matrix equations, identify the R that you want as:
For HW 02
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6. Shown: 4-point configurations for conductor resistance:
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7. HW02: Pr 2.3: Shown: 4-point configurations for Hall resistance:1 3 2
B is present I
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8. From Lec 11: how to start: q q q
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9. how to start: Landauer-Buttiker for 4-point:
q = 1, 2, 3, 4 q q q
and q = 4
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17. HW02: Pr 2.3: Shown: 4-point configurations for Hall resistance: with B:
Set one of the V’s to 0.
Choice: V3 or V4
Choose V4 since the focus is on the Hall resistance. 4 1 3 2
B out of page thinking I
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18. = 0
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19. Same matrix as p. 83 but not same electronic configuration as either (a) or (b).
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20. In solutions: “See eq’n 2.4.9”
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21. New: Switch G to Tbar description
New: Switch G to Tbar description. Why: because there is a clever experimental way to evaluate Tq p:
Typo: discuss Lec13. Roukes talks about e- steering, I think Datta tried to re-write as I steering
T2 1
B into of page
B small
B out of page
As the B-field is changed the changing transmission probability can be used to “direct” the current
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22. From the reference:
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23. left 4
forward 1 3 2
right
B present I
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24. Tforward
T2 1
Tleft
Tright
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26. Go right, left or forward defined w.r.t. the starting point:4
1  O for origin 3 2
Will consistently use:
Blue TL
Black  TF
Red  TR
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27. Go right, left or forward defined w.r.t. the starting point:
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28. Go right, left or forward defined w.r.t. the starting point:
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29. Go right, left or forward defined w.r.t. the starting point:
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30. Go right, left or forward defined w.r.t. the starting point:
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31. Go right, left or forward defined w.r.t. the starting point:
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32. So far: TO TO TO
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33. Go right, left or forward defined w.r.t. the starting point:
Remaining terms: T  TL 4
1  O for origin 3 2
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34. Go right, left or forward defined w.r.t. the starting point:
Remaining terms: T  TF 4
1  O for origin 3 2
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35. Go right, left or forward defined w.r.t. the starting point:
Remaining terms: T  TR 4
1  O for origin 3 2
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36. Go right, left or forward defined w.r.t. the starting point:
Remaining terms: T  TL 4
1  O for origin 3 2
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37. Go right, left or forward defined w.r.t. the starting point:
Remaining terms: T  TF 4
1  O for origin 3 2
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38. Go right, left or forward defined w.r.t. the starting point:
Remaining terms: T  TR 4
1  O for origin 3 2
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39. Result: Easier matrix to invert and more physically meaningful.
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40. If you check, all remaining combinations will be as shown:
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41. HW02: Pr 2.3: Shown: 4-point configurations for Hall resistance:
Previously set: V4 = 0
Which I is ideally = 0? 4 1 3 2
B is present I
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42. HW02: Pr 2.3: Shown: 4-point configurations for Hall resistance:
Previously set: V4 = 0
I2 is ideally = 0
Also: I3 = -I1 4 1 3 2
B is present I
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43. Fill this info into eq’n 2.4.9”
Goal: Find RHall associated with VHall.
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44. May be an unimportant typo in answer in some editions:
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45. Lecture 12, 08 Oct 13 In Chapters 02 and 03 in Datta:
How to correctly measure I = GV
Add scattering to Landauer-Buttiker
Sections 2.5 and 2.6: motivation: why:
2.5: Probes as scatterers especially at high bias/temps
2.6: Occupied states (q,p) as scatterers
3.1 scattering/S matrix
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46. Reflections can be introduced in several ways:
In Section 2.5:
2-t example: with broadened Fermi f0
In Section 2.6:
3-t example with reflections
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47. Landauer-Buttiker so far does not accommodate reflections:
p = 1, 2, 3, 4
q = 1, 2, 3, 4 q q q
and q = 4
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48. 2-t example: with broadened Fermi f0
In Section 2.5:
2-t example: with broadened Fermi f0
Steps to current I in Lec08:
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49. Lec08: And can fill in the kx blank for group velocity:
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50. Lec08: Calculate I:
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51. Lec08: What does ny do? It means e- has a choice of available energy levels, all of which have conductance G that become available as e- get more energy from applied V
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52. Lec08: Calculate I:
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53. Contained an assumption about the Fermi probability f0: it’s a step function O (E-Ef). This led to e-s that carry the current as shown, just above and below the Fermi level (M = 1 parabola shown here, can have more parabolas):
Also called m1
Also called m2
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54. Contained an assumption about the Fermi probability f0: it’s a step function O (E-Ef). This led to contacts and currents as shown when a potential difference V/chemical potential difference m1=m2 is present:
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55. Contained an assumption about the Fermi probability f0: it’s a step function O (E-Ef). This led to transport as shown:
F+ = m1 Ef ES
Egap
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56. But with local heating due to probes at contacts:
f0 in (b) looks like f0 in (c)
you can activate e-s in contact 2 that can populate –kx states
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57. With local heating due to probes at contacts:
There are back-currents.
Without thermal broadening of f0:
With thermal broadening of f0
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