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

2. VM Ayres, ECE802-604, F13 Lecture 04, 10 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

3. VM Ayres, ECE802-604, F13 Lecture 04, 10 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

4. VM Ayres, ECE802-604, F13 z y x -z-z y Wire up HEMT to use the triangular quantum well region in GaAs Correct for e-’s with Drain = + Note: current I is I DS n- E yE y = (-|e |)(-| E y |)

5. VM Ayres, ECE802-604, F13 Why do this: increase in Mobility in using 2-DEG region in GaAs instead of 3-DEG bulk GaAs 931C: 3D Scattering T = hot: Phonon lattice scattering T = cold: Impurity = N D +, N A - scattering Sweet spot at 300K mobility

6. VM Ayres, ECE802-604, F13 Scattering involves energy and momentum conserving interactions. Putting quantum restrictions on these interactions means that fewer can occur. Increase in Mobility is based on decrease of scattering, or said another way, increase e-s not scattered.

7. VM Ayres, ECE802-604, F13 Streetman t: Datta  m  t: The statement below is true for a group of e-s not a single scattering event.  m is an average or mean time

8. VM Ayres, ECE802-604, F13 Lecture 04, 10 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

9. VM Ayres, ECE802-604, F13 2-DEG: Major improvement in performance at low temperatures 931C: 3D Scattering T = hot: Phonon lattice scattering T = cold: Impurity = N D +, N A - scattering Sweet spot at 300K mobility

10. VM Ayres, ECE802-604, F13 2-DEG: large increase in carrier concentration n S : intrinisic

11. VM Ayres, ECE802-604, F13 2-DEG: large increase in carrier concentration n S : 3-DEG intrinisic

12. VM Ayres, ECE802-604, F13 2-DEG: Energy: Special Schrödinger eqn (Con E) that accommodates: Electronic confinement: band bending due to space charge Useful external B-field Example: ECE874, Pr. 3.5 with E-field: determine direction of motion. Datta 1.2.1 would be correct way to continue the problem to get energy levels

13. VM Ayres, ECE802-604, F13 2-DEG: Energy: 2-DEG wavefunction Use this wave function in the special Schroedinger eq’n and it will separate into k z and k x, k y parts. k z is a fixed quantized number(s). k x, k y are continuous numbers

14. VM Ayres, ECE802-604, F13 2-DEG: Energy: For the k x, k y part:

15. VM Ayres, ECE802-604, F13 Bulk Dimensionality Systems: 3-DEG Macroscopic World Bulk Materials y x z p x 2 + p y 2 + p z 2 2m* 2m* 2m* KE = Silicon Ingot Free motion in all directions p x, p y, p z can take any values B. Jacobs, PhD thesis

16. VM Ayres, ECE802-604, F13 Thin Films y x z p x 2 + p y 2 + n z 2 ħ 2  2 2m* 2m* 2m*L z 2 J.S. Moodera, Francis Bitter Magnet Lab, MIT A.K. Geim and K.S. Novoselov, Nat. Mater., 2007, 6, 183 Graphene Free motion in x and y directions Shown: Infinite potential well in z direction p z is constrained to be a number(s) Thin layers Reduced Dimensionality Systems: 2-DEG E = KE B. Jacobs, PhD thesis

17. VM Ayres, ECE802-604, F13 Carbon Nanotubes, Nanowires, Molecular Electronics y x z n x 2 ħ 2  2 + p y 2 + n z 2 ħ 2  2 2m*L x 2 2m* 2m*L z 2 Richard E. Smalley Institute, Rice University 1μm1μm Reduced Dimensionality Systems: 1-DEG Free motion in y direction Shown: Infinite potential well in x and z directions p x, p z are constrained to be a number(s) E = KE B. Jacobs, PhD thesis

18. VM Ayres, ECE802-604, F13 Quantum Dots z y x n x 2 ħ 2  2 + n y 2 ħ 2  2 + n z 2 ħ 2  2 2m*L x 2 2m*L y 2 2m*L z 2 A. Kadavanich, MRSCE, University of Wisconsin Reduced Dimensionality Systems: 0-DEG No free motion. Enter and leave QD by tunnelling Shown: Infinite potential well in x, y and z directions p x, p y, p z are constrained to be a number(s) E = B. Jacobs, PhD thesis

19. VM Ayres, ECE802-604, F13 2-DEG in a semiconductor: KE

20. VM Ayres, ECE802-604, F13 2-DEG in a semiconductor:

21. VM Ayres, ECE802-604, F13 2-DEG in a semiconductor: You have put integral travelling waves in a large box but are ignoring the edges

22. VM Ayres, ECE802-604, F13 2-DEG in a semiconductor: Standing waves in a small box. Edges matter.

23. VM Ayres, ECE802-604, F13 2-DEG in a semiconductor:

24. VM Ayres, ECE802-604, F13 2-DEG in a semiconductor: S E S is the minimum energy required for an e- to be out of a bond.

25. VM Ayres, ECE802-604, F13 2-DEG in a semiconductor: 11 Similar to: E C = E gap E S is the minimum energy required for an e- to be out of a bond.

26. VM Ayres, ECE802-604, F13 2-DEG in a semiconductor: 11

27. VM Ayres, ECE802-604, F13 2-DEG in a semiconductor: 11 kxkx Any little patch on there would have some values of k x, k y

28. VM Ayres, ECE802-604, F13 2-DEG in a semiconductor: 11 kxkx y-axis is E. The bowl is the KE that an e- has above the minimum requirement of E S required to be out of a bond p = h bar k and KE = p 2 / 2m

29. VM Ayres, ECE802-604, F13 Thin Films y x z p x 2 + p y 2 + n z 2 ħ 2  2 2m* 2m* 2m*L z 2 J.S. Moodera, Francis Bitter Magnet Lab, MIT A.K. Geim and K.S. Novoselov, Nat. Mater., 2007, 6, 183 Graphene Free motion in x and y directions Shown: Infinite potential well in z direction p z is constrained to be a number(s) Thin layers Reduced Dimensionality Systems: 2-DEG E = KE: write p in terms of h bar k

30. VM Ayres, ECE802-604, F13 Go back to this idea: You have put integral travelling waves in a large box but are ignoring the edges

31. VM Ayres, ECE802-604, F13 Combine with this idea: 11 kxkx y-axis is E. The bowl is the KE that an e- has above the minimum requirement of E S required to be out of a bond p = h bar k and KE = p 2 / 2m

32. VM Ayres, ECE802-604, F13 Count the number of available energy levels in a 2-DEG conduction band: N T (E)

33. VM Ayres, ECE802-604, F13 2-DEG in a semiconductor: N T (E)

34. VM Ayres, ECE802-604, F13 2-DEG in a semiconductor : N T (E)

35. VM Ayres, ECE802-604, F13 2-DEG in a semiconductor: N T (E)

36. VM Ayres, ECE802-604, F13 2-DEG in a semiconductor: N T (E)

37. VM Ayres, ECE802-604, F13 2-DEG in a semiconductor : N T (E)

38. VM Ayres, ECE802-604, F13 2-DEG in a semiconductor : N T (E)

39. VM Ayres, ECE802-604, F13 2-DEG in a semiconductor: N T (E)

40. VM Ayres, ECE802-604, F13 Use N T (E) to get energy density of states N(E):

41. VM Ayres, ECE802-604, F13 Your Homework Pr 1.3: 1 Deg in a semiconductor:

42. VM Ayres, ECE802-604, F13 Your Homework Pr 1.3: 1 Deg in a semiconductor:

43. VM Ayres, ECE802-604, F13

44. Use N(E) to get concentration n S

45. VM Ayres, ECE802-604, F13 Use N(E) to get concentration n S

46. VM Ayres, ECE802-604, F13

47. Fermi wavenumber k f :

48. VM Ayres, ECE802-604, F13 Corresponding Fermi velocityr v f :

49. VM Ayres, ECE802-604, F13 Characteristic mean free path length L m :

50. VM Ayres, ECE802-604, F13 Lecture 04, 10 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