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

2. Lecture 25, 26 Nov 13 Carbon Nanotubes and Graphene
CNT/Graphene electronic properties
sp2: electronic structure
2DEG: E-k relationship/graph for graphene and transport
1DEG: E-k relationship/graph for CNTs and transport
R. Saito, G. Dresselhaus and M.S. Dresselhaus
Physical Properties of Carbon Nanotubes
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3. dR = the greatest common divisor of 2m + n and 2n+ m
CNT Unit cell in green:
Ch = n a1 + m a2
|Ch| = a√n2 + m2 + mn
dt = |Ch|/p
cos q = a1 • Ch
|a1| |Ch|
T = t1 a1 + t2 a2
t1 = (2m + n)/ dR
t2 = - (2n + m) /dR
dR = the greatest common divisor of 2m + n and 2n+ m
|T| = √ 3(m2 + n2+nm)/dR = √ 3|Ch|/dR
N = | T X Ch |
| a1 x a2 |
= 2(m2 + n2+nm)/dR
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4. VM Ayres, ECE , F13

5. K1 is in same direction as Ch
Specify direction of Ch using choral angle
K2 is in same direction as T
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6. Transport: Real space: Ch Reciprocal space: K1 Real space: T
Transport: along CNT:
Along a Unit vector in the K2 direction
Can have any magnitude (hbar)k
(10,10)
(9,0)
(7,4)
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7. For an e- described as a wave:
Quantization of Energy E
is here
Standing wave: Quantization by m
in Ch / K1 direction
Travelling wave:
with an unquantized wave vector k in T/ K2 direction
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8. +
Transport ECNT is proportional to Egraphene2D  conduction energy level ECNT is proportional to the value of the transfer integral t
Conduction and valence energy levels
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9. k  hbark is in the transport direction
k  hbark is in the transport direction. Where k is relative to kx and ky depends on the nanotube (n,m)
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10. ZIGZAG: a1
Zigzag: Ch in a1 direction
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11. ZIGZAG: kx ky Example: which is the Ch direction, kx or ky?
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12. ZIGZAG: kx ky
Answer: ky
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13. Lec 24: Consider an (n, 0) zigzag CNT.
This is where the periodic boundary condition on ky comes from in:
That leaves just kx as open, MD calls it just k.
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14. ZIGZAG:
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15. ARMCHAIR: kx a1 ky Example: Which components cancel?
Which components add?
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16. ARMCHAIR: kx a1 ky Answer: Which components cancel? kx
Which components add? ky
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17. Lec 24: Consider an (n, n) armchair CNT.
This is where the periodic boundary condition on kX comes from in:
That leaves just kY as open, MD calls it just k.
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18. ARMCHAIR:
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19. ARMCHAIR:
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20. (4,2) CHIRAL: where Ch and T are:
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21. For chiral: from Lec 23:
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22. Therefore:
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23. (4,2) CHIRAL: where Ch and T are:
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24. Real space: Ch
Reciprocal space: K1
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25. (4,2) CHIRAL: where Ch and T are:
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26. (4,2) CHIRAL: where Ch and T are:
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27. Real space: Ch Reciprocal space: K1 Transport direction: Real space: T
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28. Transport in a 1-D Real space: Ch Reciprocal space: K1 Real space: T
A Unit vector in the K2 direction:
(10,10)
(9,0)
(7,4)
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29. Lec 05
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30. Lec 05:
6. Current I  q x n x vgroup
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31. Lec 06:
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32. Lec 24: What you can do with an E-k diagram:
Answer:
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33. Lec 07:
2-DEG:
1-DEG:
1-DEG:
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34. 2DEG Graphene: Conduction energy level for p*
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35. 2DEG Graphene:
N(E) E
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36. 1DEG CNT:
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Conduction energy levels

37. Specify: example: (n, 0) zigzag CNT.
You can write a periodic boundary condition on ky and substitute into eq’n That leaves just kx as open, MD calls it just k.
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38. VM Ayres, ECE , F13

39. Lec 07:
2-DEG:
1-DEG:
1-DEG:
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40. VM Ayres, ECE , F13

41. Same as Datta
Pr. 1.3
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42. Same as Datta Chp. 02 Four terminal Two terminal
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43. Same as Datta
Chp. 02
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44. Coherent:
Same as Datta
Chp. 03
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45. Incoherent:
Same as Datta
Chp. 03
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