ECE : Nanoelectronics Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University

VM Ayres, ECE , 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.

VM Ayres, ECE , 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.

VM Ayres, ECE , 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)

VM Ayres, ECE , F13 lProperties.asp C60 mean ball diameter 6.83 Å C60 ball outer diameter 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) science.org.uk/c60model.html

VM Ayres, ECE , F13 Show same idea on a SWCNT: Outer (Van Der Waals) Inner (repulsion) Mean structure d t

VM Ayres, ECE , F13 lProperties.asp C60 mean ball diameter 6.83 Å C60 ball outer diameter 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? science.org.uk/c60model.html

VM Ayres, ECE , F13 lProperties.asp C60 mean ball diameter 6.83 Å C60 ball outer diameter 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 science.org.uk/c60model.html

VM Ayres, ECE , 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.

VM Ayres, ECE , F13 C60 Guess: C70 Guess: C36

VM Ayres, ECE , F13 XZ Ke, et al, Physics Letters A 255 (1999) 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, Endcaps: ½ stable buckyball: C36

VM Ayres, ECE , 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

VM Ayres, ECE , F13 Endcaps: ½ stable buckyball: C20  smallest SWCNT inside MWCNT

VM Ayres, ECE , 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.

VM Ayres, ECE , 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, A. Beiser, Modern Physics, Chapter 13 E. Anderson, Quantum Mechanics, Chapter 7

VM Ayres, ECE , F13 diamond Benzene/graphene/CNTs/buckyballs polyacetylene acetylene alkane

VM Ayres, ECE , F13 ~

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

VM Ayres, ECE , F13 Anderson, Quantum Mechanics The corresponding  are: 3D in spherical coordinates

VM Ayres, ECE , F13 ~ So all of the directional character must come from 2p orbitals:

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

VM Ayres, ECE , F13 A carbon hybrid orbital comes from combining a 2s and a 2p:

VM Ayres, ECE , F13 sp 1 hybridization – Use orthonormality sp 1 (sp) hybridization:

VM Ayres, ECE , F13 ~ Problem without hybridization:

VM Ayres, ECE , F13 ~ sp 1 hybridization: ONE

VM Ayres, ECE , F13 ~ sp 1 hybridization: ONE Warning: watch axes, these directions have real meaning

VM Ayres, ECE , F13 ~ sp 1 hybridization: Warning: watch axes, these directions have real meaning

VM Ayres, ECE , F13

sp 1 hybridization:

VM Ayres, ECE , F13 sp 1 hybridization:

VM Ayres, ECE , F13 sp 1 hybridization:

VM Ayres, ECE , F13 sp 1 hybridization:

VM Ayres, ECE , F13 sp 1 hybridization:

VM Ayres, ECE , F13 sp 1 hybridization:

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

VM Ayres, ECE , F13 ~ sp 1 hybridization:

VM Ayres, ECE , F13 ~ sp 1 hybridization:

VM Ayres, ECE , F13 ~ sp 1 hybridization:

VM Ayres, ECE , F13 ~ Therefore acetylene is: sp 1 hybridization: