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

2. VM Ayres, ECE802-604, F13 Lecture 13, 10 Oct 13 In Chapters 02 and 03 in Datta: How to correctly measure I = GV Add scattering to Landauer-Buttiker Caveat: when Landauer-Buttiker doesn’t work When it does: Sections 2.5 and 2.6: motivation: why: 2.5: Probes as scatterers especially at high bias/temps 2.6: Occupied states (q,p) as scatterers 3.1 scattering/S matrix In Chapter 02 in Datta: How to correctly measure I = GV Landauer-Buttiker: all things equal 3-, 4-point probe experiments: set-up and read out Example: 4-point probe Pr. 2.4 set-up Example: 4-point probe R Hall Pr. 2.3: Roukes article

3. VM Ayres, ECE802-604, F13 In Section 2.5: 2-t example: with broadened Fermi f 0 In Section 2.6: 3-t example with reflections Lec12: Reflections can be introduced in several ways:

4. VM Ayres, ECE802-604, F13 Lec12: Landauer-Buttiker so far does not accommodate reflections: qqq p = 1, 2, 3, 4 q = 1, 2, 3, 4 and q = 4

5. VM Ayres, ECE802-604, F13 Landauer-Buttiker can be formulated to accommodate reflections if the transmission is coherent: e- waves do not change phase by interacting with a scatterer or with each other. Landauer-Buttiker can be made to accommodate non-coherent (phase changing) transmission. BUT Landauer-Buttiker does not accommodate channel-changing transmission. Why:

6. VM Ayres, ECE802-604, F13 Landauer-Buttiker can be formulated to accommodate reflections if the transmission is coherent: e- waves do not change phase by interacting with a scatterer or with each other.

7. VM Ayres, ECE802-604, F13 E > barrier height V 0 E < barrier height V 0 You already did this in Lec10: The e- is described as a wave. The scatterer described as a finite step. Interaction with the scatterer did not change the phase of the e- described as a wave: coherent transmission You evaluated an unambiguous transmission probability T for both cases.

8. VM Ayres, ECE802-604, F13 E > barrier height V 0 E < barrier height V 0 Result: Non-coherent transmission 2 nd scatterer For R

9. VM Ayres, ECE802-604, F13 Special Schroedinger equation is changing: are you solving it for n y = 1, or 3, or 2, or?? You’ll get a different wavefunction and E ~ k x 2 each time you change channels. Using a different wavefunction, you’d calculate a different T.

10. VM Ayres, ECE802-604, F13

11. A work-around:

12. VM Ayres, ECE802-604, F13 Landauer-Buttiker can be formulated to accommodate reflections if the transmission is coherent: e- waves do not change phase by interacting with a scatterer or with each other. Landauer-Buttiker can be made to accommodate non-coherent (phase changing) transmission. BUT Landauer-Buttiker does not accommodate channel-changing transmission (vertical flow). Why: need to get clear transmission probabilities T q p to evaluate a Landauer-Buttiker result.

13. VM Ayres, ECE802-604, F13 In Section 2.5: 2-t example: with broadened Fermi f 0 Steps to current I in Lec08:

14. VM Ayres, ECE802-604, F13 Contained an assumption about the Fermi probability f 0 : it’s a step function O (E-E f ). This led to e-s that carry the current as shown, just above and below the Fermi level (M = 1 parabola shown here, can have more parabolas): Also called  1 Also called  2

15. VM Ayres, ECE802-604, F13 Contained an assumption about the Fermi probability f 0 : it’s a step function O (E-E f ). This led to contacts and currents as shown when a potential difference V/chemical potential difference  1 =  2 is present: Fermi probabilities in this picture are shown next slide

16. VM Ayres, ECE802-604, F13

18. Contained an assumption about the Fermi probability f 0 : it’s a step function O (E-E f ). This led to transport as shown: EvEv ESES E gap E f-eq F+ =  1

19. VM Ayres, ECE802-604, F13 But with local heating due to probes at contacts: f 0 in (b) looks like f 0 in (c) you can activate e-s in contact 2 that can populate –k x states

20. VM Ayres, ECE802-604, F13 Hot:

21. VM Ayres, ECE802-604, F13 With thermal broadening of f 0 Without thermal broadening of f 0 : With local heating due to probes at contacts: There are back-currents.

22. VM Ayres, ECE802-604, F13

23. This part of I comes from e- that turned around (not in picture) at the barrier when approaching from the right. When e-transmits (next slide) it can use a different channel: T’

24. VM Ayres, ECE802-604, F13 = ?

25. VM Ayres, ECE802-604, F13

26. If T = T’, can get to Landauer-Buttiker but no reason why T should = T’. Especially if energies from probes took e- far from equilibrium. i as a function of how much energy E/what channel M the e- is in

27. VM Ayres, ECE802-604, F13 If T = T’, can get to Landauer-Buttiker: If T(E) = T(E)’, can take it outside the integral: Quasi-Fermi level Contact01 Quasi-Fermi level Contact02

28. VM Ayres, ECE802-604, F13 Can expect T = T’ at equilibrium. Consider: if energies from probes don’t take e- far from equilibrium:

29. VM Ayres, ECE802-604, F13 New useful G:

30. VM Ayres, ECE802-604, F13 Check: Good, recovered ‘cold’ result for G.

31. VM Ayres, ECE802-604, F13 Basically I = GV = G (  1 -  2 ) e that works when probes hotted things up but not too far from equilibrium

32. VM Ayres, ECE802-604, F13 Even without channel M changes, the transmission probability e- gets for each transmission depends on how much energy the e- had. Lots of possibilities so T(E) jumps a lot. New conductivity G^ will multiple the jumpiness by the part of sech 2 between  1 and  2, then sum up the combo.

33. VM Ayres, ECE802-604, F13 I = GV = G (  1 -  2 ) = Area e

34. VM Ayres, ECE802-604, F13 I = GV = G (  1 -  2 ) = Area e

35. VM Ayres, ECE802-604, F13 So it depends: 6 values of T (in red  1 –  2 box), but only 1-2 multiplied by something big (in blue box). How different are the values of these two? Summing up will also average them. An average of two close values is representative, an average of two far away values is not.

36. VM Ayres, ECE802-604, F13 If T(E) changes rapidly with energy, the “correlation energy”  c is said to be small. E T(E) 0.85 0.09 5 eV5.001 eV A very minimal change in e- energy and you are getting a different and much worse transmission probability.

37. VM Ayres, ECE802-604, F13 2-DEG

38. VM Ayres, ECE802-604, F13 Example: does the figure shown appear to meet the linear (I = G^ V) regime criteria? Criteria is:  1 -  2 << k B T FWHM shown on sech 2 is k B T

39. VM Ayres, ECE802-604, F13 Example: does the figure shown appear to meet the linear (I = G^ V) regime criteria? Criteria is:  1 -  2 << k B T FWHM shown is k B T Answer: No, they appear to be about the same (red and blue). However, part of F T (E) is low value. Comparing an ‘effective’  1 -  2 (green) maybe it’s OK.

40. VM Ayres, ECE802-604, F13 Lecture 13, 10 Oct 13 In Chapters 02 and 03 in Datta: How to correctly measure I = GV Add scattering to Landauer-Buttiker Caveat: when Landauer-Buttiker doesn’t work When it does: Sections 2.5 and 2.6: motivation: why: 2.5: Probes as scatterers especially at high bias/temps 2.6: Occupied states (q,p) as scatterers 3.1 scattering/S matrix In Chapter 02 in Datta: How to correctly measure I = GV Landauer-Buttiker: all things equal 3-, 4-point probe experiments: set-up and read out Example: 4-point probe Pr. 2.4 set-up Example: 4-point probe R Hall Pr. 2.3: Roukes article

41. VM Ayres, ECE802-604, F13 HW02: Pr. 2.4: test circuit for 1-DEG resistances in series:

42. VM Ayres, ECE802-604, F13 HW02: Pr. 2.4: test circuit for 1-DEG resistances in series: Now: B out of page. Solve this problem. Find R.

43. VM Ayres, ECE802-604, F13 Landauer-Buttiker treat places and e- can go equally to start set-up: Show some work.

44. VM Ayres, ECE802-604, F13 This is a 2-terminal circuit: R 2-t. 1 and 3 have probes, but 2 and 4 do not: they are just real estate but e- can be deflected into them by a B-field. Which of 1, 2, 3, 4, will you set to V = 0?

45. VM Ayres, ECE802-604, F13 This is a 2-terminal circuit: R 2-t. 1 and 3 have probes, but 2 and 4 do not: they are just real estate but e- can be deflected into them by a B-field. Answer: ordinary choice: V 3 = 0.

46. VM Ayres, ECE802-604, F13 V 3 = 0: 3x3 G matrix Switch to a T p q description: 3x3 T 14, … etc. matrix Switch to a T L, T R, and T F (2 choices) description.

47. VM Ayres, ECE802-604, F13 Pr. 2.3Pr. 2.4