1 / 38

ECE 802-604: Nanoelectronics

ECE 802-604: Nanoelectronics. Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University ayresv@msu.edu. Lecture 17, 24 Oct 13. Datta 3.1: scattering/S matrix Combining two 2x2 scattering matrices combine the scatterers coherently

stoupin
Télécharger la présentation

ECE 802-604: Nanoelectronics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related