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ECE 802-604: Nanoelectronics

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 sp 2 : electronic structure

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ECE 802-604: Nanoelectronics

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

  2. 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

  3. 0 VM Ayres, ECE802-604, F13

  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

  5. Graphene Unit cell: VM Ayres, ECE802-604, F13

  6. VM Ayres, ECE802-604, F13

  7. 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

  8. 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

  9. VM Ayres, ECE802-604, F13

  10. Simple way: VM Ayres, ECE802-604, F13

  11. Can guess what the terms in brackets the have to be: VM Ayres, ECE802-604, F13

  12. VM Ayres, ECE802-604, F13

  13. VM Ayres, ECE802-604, F13

  14. Why this criteria: VM Ayres, ECE802-604, F13

  15. Real space Reciprocal space Similar to <111> 0 <100> in a cubic system VM Ayres, ECE802-604, F13

  16. 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

  17. 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 Graphene versus polyacetylene: VM Ayres, ECE802-604, F13

  18. Graphene: HAB types: find the graphene factor: 60º 60º Compare with polyacetylene: Not the same VM Ayres, ECE802-604, F13

  19. Graphene factor: HAB types: VM Ayres, ECE802-604, F13

  20. Graphene factor: HAB types: VM Ayres, ECE802-604, F13

  21. Graphene factor: HAB types: VM Ayres, ECE802-604, F13

  22. Graphene factor: HAB types: VM Ayres, ECE802-604, F13

  23. Graphene factor: HAB types: VM Ayres, ECE802-604, F13

  24. Graphene factor: HAB types: VM Ayres, ECE802-604, F13

  25. Graphene factor: HAB types: VM Ayres, ECE802-604, F13

  26. Graphene factor: HAB types: f(k): VM Ayres, ECE802-604, F13

  27. Graphene: HAB types: VM Ayres, ECE802-604, F13

  28. 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

  29. Find E-k: VM Ayres, ECE802-604, F13

  30. Find E-k: VM Ayres, ECE802-604, F13

  31. Find E-k: VM Ayres, ECE802-604, F13

  32. Find E-k: VM Ayres, ECE802-604, F13

  33. 2D graphene kx and ky dependence 1D polyacetylene kx dependence This picture is a slice This picture isn’t VM Ayres, ECE802-604, F13

  34. 2D graphene kx and ky dependence Example: label the axes for this figure. VM Ayres, ECE802-604, F13

  35. 2D graphene kx and ky dependence Answer: E ky kx VM Ayres, ECE802-604, F13

  36. 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

  37. 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

  38. 30o VM Ayres, ECE802-604, F13

  39. Example: how many equivalent K points are there? How would you get to them? 30o VM Ayres, ECE802-604, F13

  40. Answer: 6 equivalent K-points E ky kx VM Ayres, ECE802-604, F13

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