ECE 802-604: Nanoelectronics Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University ayresv@msu.edu
Lecture 22, 12 Nov 13 Carbon Nanotubes and Graphene CNT/Graphene electronic properties sp2: electronic structure Find E-k relationship/graph for polyacetylene Find E-k relationship/graph for graphene R. Saito, G. Dresselhaus and M.S. Dresselhaus Physical Properties of Carbon Nanotubes VM Ayres, ECE802-604, F13
VM Ayres, ECE802-604, F13
1 Find Unit cell “a” 2 Find k: 3 Find H and S elements 4 Rules for finding the electronic structure (p. 21): 1 Find Unit cell “a” 2 Find k: 3 Find H and S elements Det [H – SI] =0 4 Solve for E(k) VM Ayres, ECE802-604, F13
Graphene Unit cell: VM Ayres, ECE802-604, F13
VM Ayres, ECE802-604, F13
Real space unit vectors and Unit cell: Area of Unit cell = a1 x a2 = same a one on the hexagons VM Ayres, ECE802-604, F13
1 Find Unit cell “a” 2 Find k: 3 Find H and S elements 4 Rules for finding the electronic structure (p. 21): 1 Find Unit cell “a” 2 Find k: 3 Find H and S elements Det [H – SI] =0 4 Solve for E(k) VM Ayres, ECE802-604, F13
VM Ayres, ECE802-604, F13
Simple way: VM Ayres, ECE802-604, F13
Can guess what the terms in brackets the have to be: VM Ayres, ECE802-604, F13
VM Ayres, ECE802-604, F13
VM Ayres, ECE802-604, F13
Why this criteria: VM Ayres, ECE802-604, F13
Similar to <111> 0 <100> in a cubic system Real space Reciprocal space Similar to <111> 0 <100> in a cubic system VM Ayres, ECE802-604, F13
1 Find Unit cell “a” 2 Find k: 3 Find H and S elements 4 Rules for finding the electronic structure (p. 21): 1 Find Unit cell “a” 2 Find k: 3 Find H and S elements Det [H – SI] =0 4 Solve for E(k) VM Ayres, ECE802-604, F13
Graphene versus polyacetylene: Similar: 2 inequivalent C-atoms Self energy HAA and HBB terms = same = e2p Self energy SAA and SBB terms = same = 1 Dissimilar: 3 nearest neighbor C-atoms at 120º not 2 nearest neighbor atoms at 180º Nearest neighbor HAB and HBA terms = same = (graphene factor) t VM Ayres, ECE802-604, F13
Graphene: HAB types: find the graphene factor: 60º 60º Compare with polyacetylene: Not the same VM Ayres, ECE802-604, F13
Graphene factor: HAB types: VM Ayres, ECE802-604, F13
Graphene factor: HAB types: VM Ayres, ECE802-604, F13
Graphene factor: HAB types: VM Ayres, ECE802-604, F13
Graphene factor: HAB types: VM Ayres, ECE802-604, F13
Graphene factor: HAB types: VM Ayres, ECE802-604, F13
Graphene factor: HAB types: VM Ayres, ECE802-604, F13
Graphene factor: HAB types: VM Ayres, ECE802-604, F13
Graphene factor: HAB types: f(k): VM Ayres, ECE802-604, F13
Graphene: HAB types: VM Ayres, ECE802-604, F13
1 Find Unit cell “a” 2 Find k: 3 Find H and S elements 4 Rules for finding the electronic structure (p. 21): 1 Find Unit cell “a” 2 Find k: 3 Find H and S elements Det [H – SI] =0 4 Solve for E(k) VM Ayres, ECE802-604, F13
Find E-k: VM Ayres, ECE802-604, F13
Find E-k: VM Ayres, ECE802-604, F13
Find E-k: VM Ayres, ECE802-604, F13
Find E-k: VM Ayres, ECE802-604, F13
2D graphene 1D polyacetylene kx and ky dependence 1D polyacetylene kx dependence This picture is a slice This picture isn’t VM Ayres, ECE802-604, F13
2D graphene kx and ky dependence Example: label the axes for this figure. VM Ayres, ECE802-604, F13
2D graphene kx and ky dependence E ky kx Answer: VM Ayres, ECE802-604, F13
Three points about E-k E-kx & ky: Smallest (best: metallic) bandgap between E+ and E- at K. Biggest (worst) bandgap between E+ and E- at G. VM Ayres, ECE802-604, F13
Three points about E-k E-kx & ky: Smallest (best) bandgap between E+ and E- at K: Example where are K? Biggest (worst) bandgap between E+ and E- at G: at kx = ky = 0 VM Ayres, ECE802-604, F13
30o VM Ayres, ECE802-604, F13
Example: how many equivalent K points are there? How would you get to them? 30o VM Ayres, ECE802-604, F13
Answer: 6 equivalent K-points ky kx VM Ayres, ECE802-604, F13