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

2. VM Ayres, ECE802-604, F13 Lecture 19, 31 Oct 13 Carbon Nanotubes and Graphene Carbon nanotube/Graphene physical structure Carbon bond hybridization is versatile : sp 1, sp 2, and sp 3 sp 2 : origin of CNT/Graphene mechanical and electronic structures CNT/Graphene electronic properties R. Saito, G. Dresselhaus and M.S. Dresselhaus Physical Properties of Carbon Nanotubes Imperial College Press, London, 1998.

3. VM Ayres, ECE802-604, F13 Lecture 19, 31 Oct 13 Carbon Nanotubes and Graphene Carbon nanotube/Graphene physical structure SWCNT endcaps Carbon bond hybridization is versatile : sp 1, sp 2, and sp 3 sp 2 : origin of mechanical and electronic structures Carbon nanotube/Graphene electronic ‘structure’ R. Saito, G. Dresselhaus and M.S. Dresselhaus Physical Properties of Carbon Nanotubes Imperial College Press, London, 1998.

4. VM Ayres, ECE802-604, F13 Lec 17: Introduction Many different types of wrapping result in a seamless cylinder. But The particular cylinder wrapping dictates the electronic and mechanical properties. Buckyball endcaps (10,10) (9,0) (7,4)

5. VM Ayres, ECE802-604, F13 http://www.sesres.com/Physica lProperties.asp C60 mean ball diameter 6.83 Å C60 ball outer diameter 10.18 Å C60 ball inner diameter 3.48 Å Endcaps: ½ stable buckyball: C60 From Wikipedia: C60 is very stable truncated icosahedron 60 vertices with a carbon atom at each vertex and 32 faces 20 hexagons and 12 pentagons where no pentagons share a vertex The van der Waals diameter of a C60 molecule is about 1.1 nanometers (nm) http://www.creative- science.org.uk/c60model.html

6. VM Ayres, ECE802-604, F13 Show same idea on a SWCNT: Outer (Van Der Waals) Inner (repulsion) Mean structure d t

7. VM Ayres, ECE802-604, F13 http://www.sesres.com/Physica lProperties.asp C60 mean ball diameter 6.83 Å C60 ball outer diameter 10.18 Å C60 ball inner diameter 3.48 Å Endcaps: ½ stable buckyball: C60 From Wikipedia: C60 is very stable truncated icosahedron 60 vertices with a carbon atom at each vertex and 32 faces 20 hexagons and 12 pentagons where no pentagons share a vertex The van der Waals diameter of a C60 molecule is about 1.1 nanometers (nm) Example: which diameter should you use as an estimate for matching a SWCNT diameter? http://www.creative- science.org.uk/c60model.html

8. VM Ayres, ECE802-604, F13 http://www.sesres.com/Physica lProperties.asp C60 mean ball diameter 6.83 Å C60 ball outer diameter 10.18 Å C60 ball inner diameter 3.48 Å Endcaps: ½ stable buckyball: C60 From Wikipedia: C60 is very stable truncated icosahedron 60 vertices with a carbon atom at each vertex and 32 faces 20 hexagons and 12 pentagons where no pentagons share a vertex The van der Waals diameter of a C60 molecule is about 1.1 nanometers (nm) Example: which diameter should you use as an estimate for matching a SWCNT diameter? Answer http://www.creative- science.org.uk/c60model.html

9. VM Ayres, ECE802-604, F13 Endcaps: ½ stable buckyball: C70 From Wikipedia: C70 is also stable 70 vertices with a carbon atom at each vertex 25 hexagons and 12 pentagons The mean ball diameter is ? LBL: The structure of C70 molecule. Red atoms indicate five hexagons additional to the C60 molecule.

10. VM Ayres, ECE802-604, F13 C60 Guess: C70 Guess: C36

11. VM Ayres, ECE802-604, F13 XZ Ke, et al, Physics Letters A 255 (1999) 294-300 http://www.lbl.gov/tt/techs/lbnl1 366.html The above image depicts the D6h isomer of the C36 molecule Piskoti C., Yarger J., Zettl A., "A New Carbon Solid", Letters to Nature, v 393, 1997. Endcaps: ½ stable buckyball: C36

12. VM Ayres, ECE802-604, F13 The smallest possible buckyball is the dodecahedral C20, a shape consisting of 12 pentagonal faces and no hexagonal faces. ~4 angstroms Endcaps: ½ stable buckyball: C20

13. VM Ayres, ECE802-604, F13 Endcaps: ½ stable buckyball: C20  smallest SWCNT inside MWCNT

14. VM Ayres, ECE802-604, F13 Example Problems: 8.Based on d t ~ 0.4 nm, what ultrasmall SWCNT structures (n.m) are possible? 9.What types of CNT are they? 10.What physical issues would be involved in the self-assembly of these SWCNTs? HW04: Find C h, |C h |, cos , , T, |T |, and N for the smallest SWCNT capped with a C20.

15. VM Ayres, ECE802-604, F13

18. Lecture 19, 31 Oct 13 Carbon Nanotubes and Graphene Carbon nanotube/Graphene physical structure Carbon bond hybridization is versatile : sp 1, sp 2, and sp 3 sp 2 : origin of CNT/Graphene mechanical and electronic structures CNT/Graphene electronic properties R. Saito, G. Dresselhaus and M.S. Dresselhaus Physical Properties of Carbon Nanotubes Imperial College Press, London, 1998. A. Beiser, Modern Physics, Chapter 13 E. Anderson, Quantum Mechanics, Chapter 7

19. VM Ayres, ECE802-604, F13 diamond Benzene/graphene/CNTs/buckyballs polyacetylene acetylene alkane

20. VM Ayres, ECE802-604, F13 ~

21. ~

22. Anderson, Quantum Mechanics These are simulations of e- |  | 2 with 1s, 2s, and 2p marked:

23. VM Ayres, ECE802-604, F13 Anderson, Quantum Mechanics The corresponding  are: 3D in spherical coordinates

24. VM Ayres, ECE802-604, F13 ~ So all of the directional character must come from 2p orbitals:

25. VM Ayres, ECE802-604, F13 Beiser, Modern Physics Boundary surface diagrams s-orbitals are all spherical: no preferred direction p-orbitals are highly directional

26. VM Ayres, ECE802-604, F13 A carbon hybrid orbital comes from combining a 2s and a 2p:

27. VM Ayres, ECE802-604, F13 sp 1 hybridization – Use orthonormality sp 1 (sp) hybridization:

28. VM Ayres, ECE802-604, F13 ~ Problem without hybridization:

29. VM Ayres, ECE802-604, F13 ~ sp 1 hybridization: ONE

30. VM Ayres, ECE802-604, F13 ~ sp 1 hybridization: ONE Warning: watch axes, these directions have real meaning

31. VM Ayres, ECE802-604, F13 ~ sp 1 hybridization: Warning: watch axes, these directions have real meaning

32. VM Ayres, ECE802-604, F13

33. sp 1 hybridization:

34. VM Ayres, ECE802-604, F13 sp 1 hybridization:

35. VM Ayres, ECE802-604, F13 sp 1 hybridization:

36. VM Ayres, ECE802-604, F13 sp 1 hybridization:

37. VM Ayres, ECE802-604, F13 sp 1 hybridization:

38. VM Ayres, ECE802-604, F13 sp 1 hybridization:

39. VM Ayres, ECE802-604, F13 sp 1 hybridization: Can get the 4 th equation for 4 unknowns from the spherical character of the |2s> components: Therefore:

40. VM Ayres, ECE802-604, F13 ~ sp 1 hybridization:

41. VM Ayres, ECE802-604, F13 ~ sp 1 hybridization:

42. VM Ayres, ECE802-604, F13 ~ sp 1 hybridization:

43. VM Ayres, ECE802-604, F13 ~ Therefore acetylene is: sp 1 hybridization: