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ECE 802-604: Nanoelectronics

Explore how scattering matrices combine scatterers coherently and incoherently to find transmission probabilities in nanoelectronics. Learn about Dresselhaus, graphene, and carbon nanotubes.

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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 17, 24 Oct 13 Datta 3.1: scattering/S matrix Combining two 2x2 scattering matrices combine the scatterers coherently combine the scatterers incoherently combine the scatterers with partial coherence Goal: find the transmission probability T through a complete structure that contains the scatterers Dresselhaus: Graphene and Carbon Nanotubes VM Ayres, ECE802-604, F13

  3. Lec 15: What is a scattering S matrix?: VM Ayres, ECE802-604, F13

  4. 1. Lec 15: 2. 3. Game plan: for each Si j element: determine: is it a reflection or a transmission? VM Ayres, ECE802-604, F13

  5. Datta, Sec. 3.1: Two propagating modes into one propagating mode: expect scattering due to occupied states Coherent Conductor VM Ayres, ECE802-604, F13

  6. Lec 15: Example: find the S-matrix for the Buttiker/Enquist diagrams shown. Incoherent Conductor VM Ayres, ECE802-604, F13

  7. Example: find the S-matrix for both of the diagrams shown. VM Ayres, ECE802-604, F13

  8. Example: find the scattering matrix S for the diagram shown within the red box: VM Ayres, ECE802-604, F13

  9. VM Ayres, ECE802-604, F13

  10. VM Ayres, ECE802-604, F13

  11. Basic form: VM Ayres, ECE802-604, F13

  12. Two points: Point 01: Reflections are not really the same. One incorporates the influence of Leads 2 and 4 and the other doesn’t. Same is true for transmissions. Therefore: Let r -> r and r’ with the influence of Leads 2 and 4 Let t -> t and t’ with the influence of Leads 2 and 4 VM Ayres, ECE802-604, F13

  13. VM Ayres, ECE802-604, F13

  14. Two points: Point 01: Reflections are not really the same. One incorporates the influence of Leads 2 and 4 and the other doesn’t. Same is true for transmissions. Therefore: Let r -> r, and r’ with the influence of Leads 2 and 4 Let t -> t, and t’ with the influence of Leads 2 and 4 VM Ayres, ECE802-604, F13

  15. VM Ayres, ECE802-604, F13

  16. Two points: Point 02: the influence of Lead 3 must be included. Lead 3 directly influences a1 and b1. Not usual a13 = a1 3 VM Ayres, ECE802-604, F13

  17. Answer: With these two points included: Not usual a13 = a1 3 VM Ayres, ECE802-604, F13

  18. Example: find the scattering matrix S for the diagram shown within the red box: Answer: With these two points included: VM Ayres, ECE802-604, F13

  19. a13 = = b24 The individual S matrices = little s(1) and little s(2) are: VM Ayres, ECE802-604, F13

  20. HW03: VA Pr. 01: Combine the two 2x2 scattering matrices given on p. 126 by eliminating a5 and b5 to obtain the S-matrix for the composite structure with the matrix elements given in eq’n 3.2.1. VM Ayres, ECE802-604, F13

  21. Could analyze any of the 4 terms. Looking at “t”: Re-write VM Ayres, ECE802-604, F13

  22. Why “t”: a13 into b24 via combined S-matrix element “t” a13 = = b24 Because it represent the goal: find the overall transmission. VM Ayres, ECE802-604, F13

  23. Goal: find the transmission probability T through a complete structure that contains the scatterers So far: found element “t” in a combined S-matrix. What is its relationship to T? VM Ayres, ECE802-604, F13

  24. 1. Lec 15: 2. 3. Game plan: for each Si j element: determine: is it a reflection or a transmission? VM Ayres, ECE802-604, F13

  25. More accurately, transmission probabilities T  t an t’ elements. You could also solve for individual reflection probabilities. VM Ayres, ECE802-604, F13

  26. VM Ayres, ECE802-604, F13

  27. HW03: VA Pr. 02: VM Ayres, ECE802-604, F13

  28. Lecture 17, 24 Oct 13 Datta 3.1: scattering/S matrix Combining two 2x2 scattering matrices combine the scatterers coherently combine the scatterers incoherently combine the scatterers with partial coherence Goal: find the transmission probability T through a complete structure that contains the scatterers Dresselhaus Graphene and Carbon Nanotubes Carbon nanotube structure Carbon bond hybridization is versatile : sp1, sp2, and sp3 Graphene VM Ayres, ECE802-604, F13

  29. CNT Structure • Introduction • The Basis Vectors: a1 and a2 • The Chiral Vector: Ch • The Chiral Angle: cos(q) • The Translation Vector: T • The Unit Cell of a CNT • Headcount of available p electrons R. Saito, G. Dresselhaus and M.S. Dresselhaus, Physical Properties of Carbon Nanotubes, Imperial College Press, London, 1998. VM Ayres, ECE802-604, F13

  30. Introduction A single wall Carbon Nanotube is a single graphene sheet wrapped into a cylinder. VM Ayres, ECE802-604, F13

  31. Introduction Buckyball endcaps Many different types of wrapping result in a seamless cylinder. But The particular cylinder wrapping dictates the electronic and mechanical properties. VM Ayres, ECE802-604, F13

  32. Introduction Example of mechanical properties: Raman Spectroscopy & phonons Light in Different wavelength light out (10, 10) SWCNT Phonons Breathing mode Tangential mode VM Ayres, ECE802-604, F13

  33. Introduction Example of mechanical properties: Raman Spectroscopy & phonons Tangential Mode Semiconducting CNT Light in Different wavelength light out Tangential Mode Metallic CNT Phonons Semiconducting & Metallic CNT Mix VM Ayres, ECE802-604, F13

  34. Introduction VM Ayres, ECE802-604, F13

  35. The Basis Vectors a1 = √3 a x + 1 a y 2 2 a2 = √3 a x - 1 a y 2 2 where magnitude a = |a1| = |a2| VM Ayres, ECE802-604, F13

  36. The Basis Vectors 1.44 Angstroms VM Ayres, ECE802-604, F13

  37. The Basis Vectors a1 = √3 a x + 1 a y 2 2 a2 = √3 a x - 1 a y 2 2 1.44 A 1.44 A 120o a Magnitude a = 2 [ (1.44 Angstroms)cos(30) ] = 2.49 Angstroms VM Ayres, ECE802-604, F13

  38. The Basis Vectors Note that the 1.44 Angstrom value is slightly different for a buckyball (0-D), a CNT (1-D) and in a graphite sheet (2-D). This is due to curvature effects. VM Ayres, ECE802-604, F13

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