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

2. VM Ayres, ECE802-604, F13 Lecture 06, 17 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 Examples

3. VM Ayres, ECE802-604, F13 Lec05: Example: write down the wave function for a 1-DEG

4. VM Ayres, ECE802-604, F13 Lec 05: Example: write down the energy eigenvalues for a 1- DEG assuming an infinite square well potential in the quantized directions

5. VM Ayres, ECE802-604, F13 Example: draw a diagram of this 1-DEG kxkx kyky kzkz x y z Width W in y Thickness t in z t W

6. VM Ayres, ECE802-604, F13 Example: write down the energy eigenvalues for a 1-DEG assuming an infinite square well potential in the quantized directions. Assume n z = 1 st and Ly  W

7. VM Ayres, ECE802-604, F13 Example: write down the energy eigenvalues for a 1-DEG assuming an infinite square well potential in the quantized directions. Assume n z = 1 st and L y  W Answer:

8. VM Ayres, ECE802-604, F13 Example: find the number of energy levels N T (E) for a 1-DEG assuming an infinite square well potential in the quantized directions

9. VM Ayres, ECE802-604, F13 Example: Generally what is the relation of the N(E) to N T (E)? Write this down for both a 2-DEG and a 1-DEG.

10. VM Ayres, ECE802-604, F13 Answer:

11. VM Ayres, ECE802-604, F13 Example: Generally what is the relation of concentration n s to N(E)?

12. VM Ayres, ECE802-604, F13 Example: Generally what is the relation of concentration n to N(E)? Answer:n = N(E) n-DEG f 0 (E) dE Key for correct nanotechnology

13. VM Ayres, ECE802-604, F13 Example: How do you define “hot” versus “cold” for the Fermi probability f 0 (E)?

14. VM Ayres, ECE802-604, F13 Answer: Cold: Hot: The definitions are what the denominator is doing: You can’t meet the Cold condition by any change in T. The only way to do it is with E f > E: Cold means the semiconductor is degenerate.

15. VM Ayres, ECE802-604, F13

16. Hint: Plot n versus (E f – E S )/ E 1 not versus E f

17. VM Ayres, ECE802-604, F13 Parabolic potential is new. Why interesting: this is the single electron transistor (SET) Kastner article, MIT

18. VM Ayres, ECE802-604, F13 2-DEG: Before: U(x,y) = 0 and A = 0: no E or B Now: U(x,y) = U(y) = ½ m  0 2 y 2. Still have A = 0: no E or B but let’s get ready for B anyway

19. VM Ayres, ECE802-604, F13 U(x,y) = U(y), and B is possible: x y z e- I DS B Like Hall effect: expect: the x motion is disturbed by the B-field

20. VM Ayres, ECE802-604, F13 2-DEG -> 1-DEG: U(y) = ½ m  0 2 y 2 B = 0Now put in: x x

21. VM Ayres, ECE802-604, F13 Wavefunction: x

22. VM Ayres, ECE802-604, F13 Energy eigenvalues are: USE this in your HW

23. VM Ayres, ECE802-604, F13 Now find N(E) Now find n L

24. VM Ayres, ECE802-604, F13 Useful B-field: experimental measures: In real life, electron densities and mobilities do not come printed on nanowires, nanotubes or graphene sheets!

25. VM Ayres, ECE802-604, F13 Useful B-field: experimental measures: What happens when you run a Hall effect measurement in a 2- DEG? Measurement set-up:

26. VM Ayres, ECE802-604, F13 Expectation: Drude model: wrong: 2-DEG:

27. VM Ayres, ECE802-604, F13 Expectation: Drude model: wrong: Write in terms of something you can measure: J:

28. VM Ayres, ECE802-604, F13 Expectation: Drude model: wrong: Dig out your resistivities and then do V = IR

29. VM Ayres, ECE802-604, F13 V Hall VxVx Dotted lines are fictitious Expectation: Drude model: wrong:

30. VM Ayres, ECE802-604, F13 Expectation: Drude model: wrong: Any low-field place where the measurement is actually doing this, life is good.

31. VM Ayres, ECE802-604, F13 Any low-field place where the measurement is actually doing this, life is good.

32. VM Ayres, ECE802-604, F13 What happens as you increase B: V Hall develops a staircase V x develops oscillations

33. VM Ayres, ECE802-604, F13 What happens when you run a Hall effect measurement in a 2- DEG? Stated without proof: The density of states used to be a constant: Now it’s a bunch (n + ½) of spikes (delta function). Each n = 0, 1, … is called a Landau level.

34. VM Ayres, ECE802-604, F13 2-DEG density of states + B-field:  E – (E S + E n )] 1 st : n = 02 nd : n = 1

35. VM Ayres, ECE802-604, F13 Spikes in N(E) => spikes in n S => spikes/troughs in current Which can be interpreted as an oscillation in resistivity. Resistivities are proportional to the measured voltages

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

37. VM Ayres, ECE802-604, F13