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

2. VM Ayres, ECE802-604, F13 Lecture 26, 02 Dec 13 Carbon Nanotubes and Graphene CNT/Graphene electronic properties sp 2 : electronic structure 2DEG: E-k relationship/graph for graphene and transport 1DEG: E-k relationship/graph for CNTs and transport Examples Molecular Electronics R. Saito, G. Dresselhaus and M.S. Dresselhaus Physical Properties of Carbon Nanotubes

3. 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 |T| = √ 3 (m 2 + n 2 +nm)/d R = √ 3|C h |/d R N = | T X C h | | a 1 x a 2 | = 2(m 2 + n 2 +nm)/d R

4. VM Ayres, ECE802-604, F13 Example: is C h for the armchair CNT at right angles to C h for the zigzag CNT?

5. VM Ayres, ECE802-604, F13 Example: is C h for the armchair CNT at right angles to C h for the zigzag CNT? Answer: No. It’s at an angle. HW: evaluate the angle.

6. VM Ayres, ECE802-604, F13 a1a1 If this is the specified unit vector system, then armchair C h is at the chiral angle and zigzag: C h in a 1 direction.

7. VM Ayres, ECE802-604, F13 Example: for the paper cutout, is C h for the armchair CNT at right angles to C h for the zigzag CNT?

8. VM Ayres, ECE802-604, F13 Example: for the paper cutout, is C h for the armchair CNT at right angles to C h for the zigzag CNT? Answer. No. Answer doesn’t change.

9. VM Ayres, ECE802-604, F13 Example: for the paper cutout, number and create the largest possible zigzag CNT

10. VM Ayres, ECE802-604, F13 Example: for the paper cutout, number and create the largest possible zigzag CNT Answer: (5,0). HW: evaluate T and cut out the proper Unit cell length.

11. VM Ayres, ECE802-604, F13 Example: for the paper cutout, number and create the largest possible armchair CNT

12. VM Ayres, ECE802-604, F13 Example: for the paper cutout, number and create the largest possible armchair CNT Answer: (3,3). HW: evaluate T and cut out the proper Unit cell length.

13. VM Ayres, ECE802-604, F13 ARMCHAIR: ZIGZAG: Example: Unit vectors a 1 and a 2 are not pointing in the same directions in (a) and (b). What is the goal of each arrangement?

14. VM Ayres, ECE802-604, F13 ARMCHAIR: ZIGZAG: Example: Unit vectors a 1 and a 2 are not pointing in the same directions in (a) and (b). What is the goal of each arrangement? Answer:

15. VM Ayres, ECE802-604, F13 Lec 24: Graphene: the 6 equivalent K-points  Bottom of the conduction band the 6 equivalent K-points  metallic E kyky kxkx This factor slices the graphene E g2D

16. VM Ayres, ECE802-604, F13 Lec 24: At a K- point = metallic: Condition: Armchair (n,n) are always metallic

17. VM Ayres, ECE802-604, F13 Lec 24: At a K- point = metallic: Condition: Example: Prove this condition. First: identify the Unit vector system being used.

18. VM Ayres, ECE802-604, F13 ARMCHAIR: Answer: First: identify the Unit vector system being used.

19. VM Ayres, ECE802-604, F13

24. C h = n a 1 + m a 2 |C h | = a√n 2 + m 2 + mn cos  = a 1 C h |a 1 | |C h | For HW:

25. VM Ayres, ECE802-604, F13 For HW: Find K 1 in this system. Show |K 1 | = 2  / |C h |

26. VM Ayres, ECE802-604, F13 Lec 06:

27. VM Ayres, ECE802-604, F13 Lec 24: What you can do with an E-k diagram: Answer:

28. VM Ayres, ECE802-604, F13 1DEG CNT: Conduction energy levels

29. VM Ayres, ECE802-604, F13 Lec 24: Consider an (n, n) armchair CNT. This is where the periodic boundary condition on k X comes from in: That leaves just k Y as open, MD calls it just k.

30. VM Ayres, ECE802-604, F13 Linearize graphene dependence around the K-point

31. VM Ayres, ECE802-604, F13 Lecture 26, 02 Dec 13 Molecular Electronics: Why not polyacetylene? or any conjugated “ene”? Examples of possibilities Actual performance Electronic (  ) structure brief review Mechanical (  ) structure brief review New: bond alteration structure Electronic result of bond alteration structure Qualitative

32. VM Ayres, ECE802-604, F13 CNTs: Electronic structure Armchair (n,n)Zigzag (3n,0)Armchair (≠3n,0)

33. VM Ayres, ECE802-604, F13 CNTs: Electronic device

34. VM Ayres, ECE802-604, F13 Graphene: Electronic structure

35. VM Ayres, ECE802-604, F13 Graphene: Electronic device

36. VM Ayres, ECE802-604, F13 Polyacetylene: Electronic structure

37. VM Ayres, ECE802-604, F13 Polyacetylene: Electronic device

38. VM Ayres, ECE802-604, F13 Polyphenylene: Electronic structure:

39. VM Ayres, ECE802-604, F13 Polyphenylene: Electronic device

40. VM Ayres, ECE802-604, F13

42. If it looked the same in 2008 as in 1992, there are some problems that people are still trying to solve!

43. VM Ayres, ECE802-604, F13 Actual performance: Polyphenylene and Polyactetylene Slow Variable Expected performance: Polyphenylene and Polyactetylene Quasi-ballistic like graphene and SWCNTs

44. VM Ayres, ECE802-604, F13 Factors that affect transport: Availability of electrons AND empty states to take them Scattering: – Particle-like: L  < L m < L – Wavelike (ballistic): L < L m < L  – Electrons in a 2D or 1D structure are wavelike and therefore should have limited scattering Transport mechanism: – Diffusion – Tunnelling – Ballistic (Plasmon) – Charge transfer – Soliton (Polaron) – Exciton – Hopping Injection (Contacts)

45. VM Ayres, ECE802-604, F13 Lecture 27, 03 Dec 13 Molecular Electronics: Why not polyacetylene? or any conjugated “ene”? Examples of possibilities Actual performance Electronic (  ) structure brief review Mechanical (  ) structure brief review New: bond alteration structure Electronic result of bond alteration structure Qualitative Quantitative Solitons (polarons): Su-Schreiffer-Heeger (SSH) model

46. VM Ayres, ECE802-604, F13 Division of structural and electronic properties in sp 2 makes both good: -C=C- Electronic:  -bonds Structure:  -bonds

47. VM Ayres, ECE802-604, F13 Electronic:  -bonds Structure:  -bonds Electronic: Delocalized  e-  * -conduction band e- ECE, PHY -anti-bonding e- CHM  -valence band e- ECE, PHY -bonding e- CHM

48. VM Ayres, ECE802-604, F13 -C=C- Electronic:  -bonds Structure:  -bonds MECHANICAL ELECTRICAL Division of structural and electronic properties in sp 2 makes both good:

49. VM Ayres, ECE802-604, F13 Lecture 26, 02 Dec 13 Molecular Electronics: Why not polyacetylene? or any conjugated “ene”? Examples of possibilities Actual performance Electronic (  ) structure brief review Mechanical (  ) structure brief review New: bond alteration structure Electronic result of bond alteration structure Qualitative Quantitative Solitons (polarons): Su-Schreiffer-Heeger (SSH) model

50. VM Ayres, ECE802-604, F13 Review: Polyacetylene: H AA types: “A” c ccc c H H H H H “B” c H c H -a+a c H c H

51. VM Ayres, ECE802-604, F13 New: Bond alteration polyacetylene: H AA types: No formula changes due to long single and short double bonds “A” c H c H c H c H c H “B” c H c H -a+a c H c H

52. VM Ayres, ECE802-604, F13 Review: Polyacetylene: H AB types: “A” c ccc c H H H H H “B” a c H c H a -a/2+a/2

53. VM Ayres, ECE802-604, F13 New: Bond alteration polyacetylene: H AB types “A” c H c H c H c H c H “B” c H c H -a+a c H c H “B”

54. VM Ayres, ECE802-604, F13 Also: Two “identical” bond alterations

55. VM Ayres, ECE802-604, F13 This is handled by a perturbation approach. lessmore

56. VM Ayres, ECE802-604, F13