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ECE 802-604: Nanoelectronics

Learn about evaluating transmission/reflection probabilities for scatterers in nanoelectronics, Landauer-Buttiker equation, and resistance modeling in this insightful lecture by Prof. Virginia Ayres.

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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 11, 03 Oct 13 In Chapter 02 in Datta: Transport: current I = GV V = IR => I = GV Velocity Energy levels M  M(E) Conductance G = GC in a 1-DEG Example Pr. 2.1: 2-DEG-1-DEG-2-DEG Example: 3-DEG-1-DEG-3-DEG Transmission probability: the new ‘resistance’ How to evaluate the Transmission/Reflection probability T(N) for multiple scatterers T(L) in terms of a “how far do you get length” L0 How to correctly measure I = GV Landauer-Buttiker: all things equal 3-, 4-point probe experiments: set-up and read out VM Ayres, ECE802-604, F13

  3. Lec10: Scattering: Landauer formula for G VM Ayres, ECE802-604, F13

  4. Lec10: Scattering: Landauer formula for R VM Ayres, ECE802-604, F13 … + ‘wire’ resistance: Dresselhaus

  5. Lec10: Modelled the scatterer X as a finite step potential in a certain region. Dresselhaus p. 144: static scattering, scattering by a potential in one dimension Modelled the wavelike e- as having energy E > or < V0 and got a transmission probability T E > barrier height V0 E < barrier height V0 VM Ayres, ECE802-604, F13

  6. Point01: The phase and amplitude at electrode 2 can be obtained from the phase and amplitude at electrode 1 E > V0 Phase and amplitude are the same E< V0 Amplitude is reduced but phase is the same. m1 m2 E > V0 E < V0 Phase: e- as wave is same at both contacts. This is the origin of the unchanged contact resistance GC-1 = RC VM Ayres, ECE802-604, F13

  7. Point02: Everywhere: Transport is by occupying a discreet energy level, “channel”. Say that there are M channels in the 1-DEG. Datta Dresselhaus VM Ayres, ECE802-604, F13

  8. Point02: Everywhere: Transport is by occupying a discreet energy level, “channel”. Say that there are M channels in the 1-DEG. T is the transmission probability for a channel to go from electrode 1 to electrode 2, which is given by the sum of the sum of the transmission probability from the ith to the jth channel VM Ayres, ECE802-604, F13

  9. Point02: Everywhere: Transport is by occupying a discreet energy level, “channel”. Say that there are M channels in the 1-DEG. T is the transmission probability for a channel to go from electrode 1 to electrode 2, which is given by the sum of the sum of the transmission probability from the ith to the jth channel Clearly this e- didn’t make it. Reflections are the cause of resistance in the wire. … + ‘wire’ resistance: Dresselhaus VM Ayres, ECE802-604, F13

  10. Point02: Everywhere: Transport is by occupying a discreet energy level, “channel”. Say that there are M channels in the 1-DEG. T is the transmission probability for a channel to go from electrode 1 to electrode 2, which is given by the sum of the sum of the transmission probability from the ith to the jth channel Clearly this e- didn’t make it. Reflections are the cause of resistance in the wire. But don’t rule out scattering forward again. Versus VM Ayres, ECE802-604, F13

  11. Lecture 11, 03 Oct 13 In Chapter 02 in Datta: Transport: current I = GV V = IR => I = GV Velocity Energy levels M  M(E) Conductance G = GC in a 1-DEG Example Pr. 2.1: 2-DEG-1-DEG-2-DEG Example: 3-DEG-1-DEG-3-DEG Transmission probability: the new ‘resistance’ How to evaluate the Transmission/Reflection probability T(N) for multiple scatterers T(L) in terms of a “how far do you get length” L0 How to correctly measure I = GV Landauer-Buttiker: all things equal 3-, 4-point probe experiments: set-up and read out VM Ayres, ECE802-604, F13

  12. This is a representation of 2 scatterers.Go from 2 scatterers to N scatterers per unit length L.What happens to the transmission probability T: R2 R1 VM Ayres, ECE802-604, F13

  13. Transmission probability for 2 scatterers: T = T12: Include the forward reflections that rein forces the transmission: 1 back-forth then out 2 back-forth then out VM Ayres, ECE802-604, F13

  14. Transmission probability for 2 scatterers: T = T12: Eliminate R1 and R2: VM Ayres, ECE802-604, F13

  15. Transmission probability for 2 scatterers: T = T12: Ratio the Reflection to the Transmission probability: VM Ayres, ECE802-604, F13

  16. Transmission probability for 2 scatterers: T = T12: That’s interesting. That Ratio is additive: Assuming that the scatterers are identical: VM Ayres, ECE802-604, F13

  17. Transmission probability for N identical scatterers: You’ll get the same result for N identical scatterers: VM Ayres, ECE802-604, F13

  18. Transmission probability for N identical scatterers: Now solve for transmission probability T(N): VM Ayres, ECE802-604, F13

  19. Transmission probability T(N) => T(L): Re-write N in terms of n = N/L: Define a “how far do you get length” L0: L0 is similar to a mean free path Lm but in terms of T versus lots/few R VM Ayres, ECE802-604, F13

  20. Transmission probability T(N) => T(L): Therefore: with: VM Ayres, ECE802-604, F13

  21. Transmission probability T(N) => T(L): VM Ayres, ECE802-604, F13

  22. Lecture 11, 03 Oct 13 In Chapter 02 in Datta: Transport: current I = GV V = IR => I = GV Velocity Energy levels M  M(E) Conductance G = GC in a 1-DEG Example Pr. 2.1: 2-DEG-1-DEG-2-DEG Example: 3-DEG-1-DEG-3-DEG Transmission probability: the new ‘resistance’ How to evaluate the Transmission/Reflection probability T(N) for multiple scatterers T(L) in terms of a “how far do you get length” L0 How to correctly measure I = GV Landauer-Buttiker: all things equal 3-, 4-point probe experiments: set-up and read out VM Ayres, ECE802-604, F13

  23. Now relate the transmission probability T(N) or T(L) to an e- current I using Landauer’s formula for G: VM Ayres, ECE802-604, F13

  24. Add probes: 3-point configurations: VM Ayres, ECE802-604, F13

  25. Add probes: 4-point configurations: VM Ayres, ECE802-604, F13

  26. Landauer-Buttiker formalism: Very briefly: why: h VM Ayres, ECE802-604, F13

  27. Landauer-Buttiker formalism: Very briefly: why: 3/ How close is your probe to a scatterer: e- wave interference  zero reading? 2/ Due to a physical bending, probe P1 couples preferentially to -kx states, while probe P2 couples preferentially to +kx states. You could read normal, very low or negative resistance depending on T. VM Ayres, ECE802-604, F13

  28. Landauer-Buttiker formalism: Very briefly: why: VM Ayres, ECE802-604, F13

  29. Landauer-Buttiker formalism: how: q <- p is from p to q, where p and q are any of the terminals. VM Ayres, ECE802-604, F13

  30. Landauer-Buttiker formalism: how: VM Ayres, ECE802-604, F13

  31. Landauer-Buttiker formalism: how: VM Ayres, ECE802-604, F13

  32. Landauer-Buttiker formalism: how: VM Ayres, ECE802-604, F13

  33. Landauer-Buttiker formalism: how: GOAL: Find resistance R VM Ayres, ECE802-604, F13

  34. Example: 3 point probe-aFind R for this circuit! VM Ayres, ECE802-604, F13

  35. q q q VM Ayres, ECE802-604, F13

  36. q q q VM Ayres, ECE802-604, F13

  37. q q q VM Ayres, ECE802-604, F13

  38. Landauer-Buttiker set-up is complete VM Ayres, ECE802-604, F13

  39. VM Ayres, ECE802-604, F13

  40. VM Ayres, ECE802-604, F13

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  43. VM Ayres, ECE802-604, F13

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  45. VM Ayres, ECE802-604, F13

  46. VM Ayres, ECE802-604, F13

  47. In V1 eq’n ? VM Ayres, ECE802-604, F13

  48. VM Ayres, ECE802-604, F13

  49. VM Ayres, ECE802-604, F13

  50. VM Ayres, ECE802-604, F13

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