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

2. VM Ayres, ECE802-604, F13 Lecture 23, 19 Nov 13 Carbon Nanotubes and Graphene CNT/Graphene electronic properties sp 2 : electronic structure E-k relationship/graph for polyacetylene E-k relationship/graph for graphene E-k relationship/graph for CNTs R. Saito, G. Dresselhaus and M.S. Dresselhaus Physical Properties of Carbon Nanotubes

3. VM Ayres, ECE802-604, F13 Polyacetylene E-k: E kxkx -k x

4. VM Ayres, ECE802-604, F13 To really finish: Need to model the wavefunctions: Could let  = |2p x > Could let  = |  > (  = |sp 2 > is the  -bond) ECE 802-604: Use this result, p.24: t = -1.0 s = +0.2  2p = 0.0

5. VM Ayres, ECE802-604, F13 Real space

6. VM Ayres, ECE802-604, F13 Two different spring constants: tighter k 1 (double bond) and looser k 2 single bond k1k1 k2k2

7. VM Ayres, ECE802-604, F13 Graphene E-k:

8. VM Ayres, ECE802-604, F13 To really finish: Need to model the wavefunctions: Could let  = |2p x > Could let  = |  > (  = |sp 2 > is the  -bond) ECE 802-604: Use this result, p.27: t = -3.033 Units s = 0.129 Units  2p = 0.0 Units  2p A Example: what are the Units?

9. VM Ayres, ECE802-604, F13 To really finish: Need to model the wavefunctions: Could let  = |2p x > Could let  = |  > (  = |sp 2 > is the  -bond) ECE 802-604: Use this result, p.27: t = -3.033 eV s = 0.129 pure number  2p = 0.0 eV  2p A Answer:

10. VM Ayres, ECE802-604, F13 Graphene E-k:

11. VM Ayres, ECE802-604, F13 Bottom of the conduction band: the 6 equivalent K-points E kyky kxkx

12. VM Ayres, ECE802-604, F13 What you can do with an E-k diagram: Example: What is “k” in 2D? In 1D?

13. VM Ayres, ECE802-604, F13 What you can do with an E-k diagram: Answer:

14. VM Ayres, ECE802-604, F13 Lecture 23 & 24, 19 Nov 13 Carbon Nanotubes and Graphene CNT/Graphene electronic properties sp 2 : electronic structure E-k relationship/graph for polyacetylene E-k relationship/graph for graphene E-k relationship/graph for CNTs R. Saito, G. Dresselhaus and M.S. Dresselhaus Physical Properties of Carbon Nanotubes

15. VM Ayres, ECE802-604, F13 Graphene: the 6 equivalent K-points  Bottom of the conduction band the 6 equivalent K-points  metallic E kyky kxkx Therefore: CNTs are metallic at the conditions for the K-points of graphene

16. VM Ayres, ECE802-604, F13 2 1 3 4 Use to find k Find k Find H AA, H AB, S AA, S AB Find det| H – E S | = 0 => E = E(k) Plot E versus k Rules for finding the electronic structure (p. 21):

17. VM Ayres, ECE802-604, F13 sp 2 electronic structure: CNTs – Real space (Unit cell) – Reciprocal space – Use Real and Reciprocal space to find E

18. VM Ayres, ECE802-604, F13 CNT Unit cell in green: C h = n a 1 + m a 2 |C h | = a√n 2 + m 2 + mn d t = |C h |/  cos  = a 1 C h |a 1 | |C h | T = t 1 a 1 + t 2 a 2 t 1 = (2m + n)/ d R t 2 = - (2n + m) /d R d R = the greatest common divisor of 2m + n and 2n+ m N = | T X C h | | a 1 x a 2 | = 2(m 2 + n 2 +nm)/d R

19. VM Ayres, ECE802-604, F13 Example: Evaluate K 1 for a (4,2) CNT:

20. VM Ayres, ECE802-604, F13 In class:

21. VM Ayres, ECE802-604, F13 In class:

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24. Example: Evaluate K 2 for a (4,2) CNT:

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28. Example: add a set of axes

29. VM Ayres, ECE802-604, F13 Answer: kxkx kyky

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31. 0 through 27  28 of these:

32. VM Ayres, ECE802-604, F13 E CNT is quantized

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34. Example: For a (4,2) CNT evaluate: C h,|C h|, T, |T|, K 1, K 2, |K 1 |, |K 2 |

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40. Example: Compare |b 1|, |b 2 |, with |K 1 |, |K 2 | for a (4, 2) CNT

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43. CNT E-k; Energy dispersion relations (E vs k curves): Quantization of Energy E is here K 1 is quantized by  in C h direction K 2 = k is continuous in T direction