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ECE 802-604: Nanoelectronics

ECE 802-604: Nanoelectronics. Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University ayresv@msu.edu. Lecture 06, 17 Sep 13. In Chapter 01 in Datta: Two dimensional electron gas (2-DEG) DEG goes down, mobility goes up Define mobility

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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 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 tm Count carriers nS available for current – Pr. 1.3 (1-DEG) How nS 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 VM Ayres, ECE802-604, F13

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

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

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

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

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

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

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

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

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

  12. Example: Generally what is the relation of concentration n to N(E)? Answer: n = N(E)n-DEG f0(E) dE Key for correct nanotechnology VM Ayres, ECE802-604, F13

  13. Example: How do you define “hot” versus “cold” for the Fermi probability f0(E)? VM Ayres, ECE802-604, F13

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

  15. VM Ayres, ECE802-604, F13

  16. Hint: Plot n versus (Ef – ES )/ E1 not versus Ef VM Ayres, ECE802-604, F13

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

  18. 2-DEG: Before: U(x,y) = 0 and A = 0: no E or B Now: U(x,y) = U(y) = ½ mw02y2 . Still have A = 0: no E or B but let’s get ready for B anyway VM Ayres, ECE802-604, F13

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

  20. 2-DEG -> 1-DEG: x x Now put in: B = 0 U(y) = ½ mw02y2 VM Ayres, ECE802-604, F13

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

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

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

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

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

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

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

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

  29. Expectation: Drude model: wrong: VHall Vx Dotted lines are fictitious VM Ayres, ECE802-604, F13

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

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

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

  33. 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. VM Ayres, ECE802-604, F13

  34. 2-DEG density of states + B-field: d[E – (ES + En)] 2nd: n = 1 1st: n = 0 VM Ayres, ECE802-604, F13

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

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

  37. VM Ayres, ECE802-604, F13

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