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

2. VM Ayres, ECE802-604, F13 Lecture 07, 19 Sep 13 In Chapter 01 in Datta: Two dimensional electron gas (2-DEG) DEG goes down, mobility goes up Define mobility Proportional to momentum relaxation time  m Count carriers n S available for current – Pr. 1.3 (1-DEG) How n S influences scattering in unexpected ways – Pr 1.1 (2-DEG) One dimensional electron gas (1-DEG) Special Schrödinger eqn (Con E) that accommodates: Electronic confinement: band bending due to space charge Useful external B-field Experimental measure for mobility Finish Chp. 01

3. VM Ayres, ECE802-604, F13 Given: z-dimension has a quantum confinement with widely separated energy levels such that n z = 1 st always is a good assumption. Example: if z-confinement can be modelled as an infinite potential well:

4. VM Ayres, ECE802-604, F13 2-DEG: 1-DEG:

5. VM Ayres, ECE802-604, F13 2-DEG:

6. VM Ayres, ECE802-604, F13 1-DEG: n = 0, 1, 2..

7. VM Ayres, ECE802-604, F13 1-DEG: n = 0, 1, 2.. Electric (confinement) sub-bands in units of E 1 Electric (confinement) sub-bands in units of h bar  0

8. VM Ayres, ECE802-604, F13 2-DEG: U(x,y) = 0, A = 0 1-DEG: U(x,y) = U(y) ≠ 0: hardwall, A = 0 1-DEG: U(x,y) = U(y) ≠ 0: parabolic, A = 0 “electric”  U(y)

9. VM Ayres, ECE802-604, F13 Lec 06: Add useful B-field to 2-DEG:

10. VM Ayres, ECE802-604, F13 Lec06: High B-field measurement of carrier density: number of occupied Landau levels Changes by 1 between any two levels B 1, etc. measured at maxima so there is a ‘rest’ of the oscillation Lec06: Spikes in N(E) => spikes in n S => spikes/troughs in current What you can do with a trough: use a value of |B| to turn the current OFF: magnetic switch

11. VM Ayres, ECE802-604, F13 Lec06: High B-field measurement of carrier density: number of occupied Landau levels Changes by 1 between any two levels

12. VM Ayres, ECE802-604, F13 Lec06: High B-field measurement of carrier density: number of occupied Landau levels Changes by 1 between any two levels B 1, etc. measured at maxima so there is a ‘rest’ of the oscillation Lec06: Spikes in N(E) => spikes in n S => spikes/troughs in current What you could potentially do with a trough: use a value of |B| to turn the current OFF: magnetic switch

13. VM Ayres, ECE802-604, F13 Given: z-dimension has a quantum confinement with widely separated energy levels such that n z = 1 st always is a good assumption. Example: if z-confinement can be modelled as an infinite potential well:

14. VM Ayres, ECE802-604, F13 2-DEG: U(x,y) = 0, A = 0 Start here: Now add B-field: 2-DEG: U(x,y) = 0, A ≠ 0

15. VM Ayres, ECE802-604, F13 U(y) = 0

16. VM Ayres, ECE802-604, F13 x x You pulled B out in front but in terms of a familiar function of B: the cyclotron frequency  c

17. VM Ayres, ECE802-604, F13 Compare just the mathematical form to 1-DEG parabolic electronic confinement form: 1-DEG: U(x,y) = U(y) ≠ 0: parabolic, A = 0 2-DEG: U(x,y) = 0, A ≠ 0 x

18. VM Ayres, ECE802-604, F13 Compare just the mathematical form to 1-DEG parabolic electronic confinement form: 1-DEG: U(x,y) = U(y) ≠ 0: parabolic, A = 0 2-DEG: U(x,y) = 0, A ≠ 0 x

19. VM Ayres, ECE802-604, F13 Can write down the wavefunction and energy eigenvalues by comparison to 1-DEG parabolic:

20. VM Ayres, ECE802-604, F13 Compare just the mathematical form to 1-DEG parabolic electronic confinement form: 1-DEG: U(x,y) = U(y) ≠ 0: parabolic, A = 0 2-DEG: U(x,y) = 0, A ≠ 0 x Identical

21. VM Ayres, ECE802-604, F13 Compare just the mathematical form to 1-DEG parabolic electronic confinement form: 1-DEG: U(x,y) = U(y) ≠ 0: parabolic, A = 0 2-DEG: U(x,y) = 0, A ≠ 0 x Motion in current transport direction along k x NO motion in current transport direction along k x

22. VM Ayres, ECE802-604, F13

23. 1-DEG: U(x,y) = U(y) ≠ 0: parabolic, A = 0 x Motion in current transport direction along k x n = 0, 1, 2.. x x

24. VM Ayres, ECE802-604, F13 0 1 2 x

25. 0 1 2 x x

26. y =

27. VM Ayres, ECE802-604, F13 x x However: no explicit k x dependence but there is an implicit one:

28. VM Ayres, ECE802-604, F13 kyky kxkx y x

29. Magnetic switch in 2-DEG is not working out: Spikes in N(E) => spikes in n S => spikes/troughs in current What you could do with a trough: use a value of |B| to turn the current OFF: magnetic switch No group velocity => no current. Lumps in n S but no current.

30. VM Ayres, ECE802-604, F13 Try a Magnetic switch in 1-DEG (this does work): 1-DEG: U(x,y) = U(y) ≠ 0: parabolic, A ≠ 0: B-field out of page:

31. VM Ayres, ECE802-604, F13 Mathematically similar type: parabolic (y + something) 2 E S + p y 2 /2m terms AND a y k 2  k x 2 term!

32. VM Ayres, ECE802-604, F13 Wavefunction: x Energy eigenvalues:

33. VM Ayres, ECE802-604, F13 group 0 1 2 The bigger the B-field (  c part) compared with the electronic confinement  0 part), the flatter that E-k x diagram. You can control the group velocity.

34. VM Ayres, ECE802-604, F13 Even more effective: you can control how many, even if, those bands are occupied. Magnetic switch in 1-DEG: Pr. 1.4 OFF

35. VM Ayres, ECE802-604, F13

36. Fermi level is invariant throughout device: E f2-Deg = E f1-DEG = E f

37. VM Ayres, ECE802-604, F13 First find:

38. VM Ayres, ECE802-604, F13 Now find B-values that will make these 4 levels inaccessible one by one:

39. VM Ayres, ECE802-604, F13