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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 23, 19 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 23, 19 Nov 13 Carbon Nanotubes and Graphene CNT/Graphene electronic properties sp2: electronic structure E-k relationship/graph for polyacetylene E-k relationship/graph for graphene E-k relationship/graph for CNTs R. Saito, G. Dresselhaus and M.S. Dresselhaus Physical Properties of Carbon Nanotubes VM Ayres, ECE802-604, F13

  3. Polyacetylene E-k: E -kx kx VM Ayres, ECE802-604, F13

  4. To really finish: Need to model the wavefunctions: Could let f = |2px> Could let f = |p> (f = |sp2> is the s-bond) ECE 802-604: Use this result, p.24: t = -1.0 s = +0.2 e2p = 0.0 VM Ayres, ECE802-604, F13

  5. Real space VM Ayres, ECE802-604, F13

  6. Two different spring constants:tighter k1 (double bond) and looser k2 single bond k1 k2 VM Ayres, ECE802-604, F13

  7. Graphene E-k: VM Ayres, ECE802-604, F13

  8. e2p A To really finish: Need to model the wavefunctions: Could let f = |2px> Could let f = |p> (f = |sp2> is the s-bond) ECE 802-604: Use this result, p.27: t = -3.033 Units s = 0.129 Units e2p = 0.0 Units Example: what are the Units? VM Ayres, ECE802-604, F13

  9. e2p A To really finish: Need to model the wavefunctions: Could let f = |2px> Could let f = |p> (f = |sp2> is the s-bond) Answer: ECE 802-604: Use this result, p.27: t = -3.033 eV s = 0.129 pure number e2p = 0.0 eV VM Ayres, ECE802-604, F13

  10. Graphene E-k: VM Ayres, ECE802-604, F13

  11. Bottom of the conduction band: the 6 equivalent K-points E ky kx VM Ayres, ECE802-604, F13

  12. What you can do with an E-k diagram: Example: What is “k” in 2D? In 1D? VM Ayres, ECE802-604, F13

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

  14. Lecture 23 & 24, 19 Nov 13 Carbon Nanotubes and Graphene CNT/Graphene electronic properties sp2: electronic structure E-k relationship/graph for polyacetylene E-k relationship/graph for graphene E-k relationship/graph for CNTs R. Saito, G. Dresselhaus and M.S. Dresselhaus Physical Properties of Carbon Nanotubes VM Ayres, ECE802-604, F13

  15. Graphene: the 6 equivalent K-points Bottom of the conduction band the 6 equivalent K-points metallic E ky kx Therefore: CNTs are metallic at the conditions for the K-points of graphene VM Ayres, ECE802-604, F13

  16. Rules for finding the electronic structure (p. 21): 1 Use to find k 2 Find k 3 Find HAA, HAB, SAA, SAB Find det|H – ES| = 0 => E = E(k) 4 Plot E versus k VM Ayres, ECE802-604, F13

  17. sp2 electronic structure: CNTs • Real space (Unit cell) • Reciprocal space • Use Real and Reciprocal space to find E VM Ayres, ECE802-604, F13

  18. CNT Unit cell in green: Ch = n a1 + m a2 |Ch| = a√n2 + m2 + mn dt = |Ch|/p cos q = a1 • Ch |a1| |Ch| T = t1a1 + t2a2 t1 = (2m + n)/ dR t2 = - (2n + m) /dR dR = the greatest common divisor of 2m + n and 2n+ m N = | T X Ch | | a1xa2 | = 2(m2 + n2+nm)/dR VM Ayres, ECE802-604, F13

  19. Example: Evaluate K1 for a (4,2) CNT: VM Ayres, ECE802-604, F13

  20. In class: VM Ayres, ECE802-604, F13

  21. In class: VM Ayres, ECE802-604, F13

  22. VM Ayres, ECE802-604, F13

  23. VM Ayres, ECE802-604, F13

  24. Example: Evaluate K2 for a (4,2) CNT: VM Ayres, ECE802-604, F13

  25. VM Ayres, ECE802-604, F13

  26. VM Ayres, ECE802-604, F13

  27. VM Ayres, ECE802-604, F13

  28. Example: add a set of axes VM Ayres, ECE802-604, F13

  29. Answer: ky VM Ayres, ECE802-604, F13 kx

  30. VM Ayres, ECE802-604, F13

  31. 0 through 27  28 of these: VM Ayres, ECE802-604, F13

  32. ECNT is quantized VM Ayres, ECE802-604, F13

  33. VM Ayres, ECE802-604, F13

  34. Example: For a (4,2) CNTevaluate: Ch,|Ch|, T, |T|, K1, K2, |K1|, |K2| VM Ayres, ECE802-604, F13

  35. VM Ayres, ECE802-604, F13

  36. VM Ayres, ECE802-604, F13

  37. VM Ayres, ECE802-604, F13

  38. VM Ayres, ECE802-604, F13

  39. VM Ayres, ECE802-604, F13

  40. Example: Compare |b1|, |b2|, with |K1|, |K2| for a (4, 2) CNT VM Ayres, ECE802-604, F13

  41. VM Ayres, ECE802-604, F13

  42. VM Ayres, ECE802-604, F13

  43. CNT E-k; Energy dispersion relations (E vs k curves): Quantization of Energy E is here K1 is quantized by m in Ch direction K2 = k is continuous in T direction VM Ayres, ECE802-604, F13

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