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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 04, 10 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 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 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 VM Ayres, ECE802-604, F13

  3. Lecture 04, 10 Sep 13 In Chapter 01 in Datta: Two dimensional electron gas (2-DEG) DEG goes down, mobility goes up Define mobility m 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 VM Ayres, ECE802-604, F13

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

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

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

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

  8. Lecture 04, 10 Sep 13 In Chapter 01 in Datta: Two dimensional electron gas (2-DEG) DEG goes down, mobility goes up Define mobility m 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 VM Ayres, ECE802-604, F13

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

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

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

  12. 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 VM Ayres, ECE802-604, F13

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

  14. 2-DEG: Energy: For the kx, ky part: VM Ayres, ECE802-604, F13

  15. z y x Bulk Dimensionality Systems: 3-DEG px2 + py2 + pz2 2m* 2m* 2m* KE = Free motion in all directions px , py , pz can take any values Macroscopic World Bulk Materials Silicon Ingot B. Jacobs, PhD thesis VM Ayres, ECE802-604, F13

  16. Reduced Dimensionality Systems: 2-DEG KE z px2 + py2 + nz2 ħ2p2 2m* 2m* 2m*Lz2 E = y Free motion in x and y directions Shown: Infinite potential well in z direction pz is constrained to be a number(s) x Graphene Thin Films Thin layers J.S. Moodera, Francis Bitter Magnet Lab, MIT A.K. Geim and K.S. Novoselov, Nat. Mater., 2007, 6, 183 B. Jacobs, PhD thesis VM Ayres, ECE802-604, F13

  17. z y x Reduced Dimensionality Systems: 1-DEG KE nx2 ħ2p2 + py2 + nz2 ħ2p2 2m*Lx2 2m* 2m*Lz2 E = Free motion in y direction Shown: Infinite potential well in x and z directions px , pz are constrained to be a number(s) Carbon Nanotubes, Nanowires, Molecular Electronics 1μm Richard E. Smalley Institute, Rice University B. Jacobs, PhD thesis VM Ayres, ECE802-604, F13

  18. nx2 ħ2p2 + ny2 ħ2p2 + nz2 ħ2p2 2m*Lx2 2m*Ly2 2m*Lz2 Reduced Dimensionality Systems: 0-DEG E = z No free motion. Enter and leave QD by tunnelling Shown: Infinite potential well in x, y and z directions px, py,pz are constrained to be a number(s) y x Quantum Dots A. Kadavanich, MRSCE, University of Wisconsin B. Jacobs, PhD thesis VM Ayres, ECE802-604, F13

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

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

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

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

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

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

  25. 2-DEG in a semiconductor: Similar to: e1 EC = Egap ES is the minimum energy required for an e- to be out of a bond. VM Ayres, ECE802-604, F13

  26. 2-DEG in a semiconductor: e1 VM Ayres, ECE802-604, F13

  27. 2-DEG in a semiconductor: e1 kx Any little patch on there would have some values of kx, ky VM Ayres, ECE802-604, F13

  28. 2-DEG in a semiconductor: e1 kx y-axis is E. The bowl is the KE that an e- has above the minimum requirement of ES required to be out of a bond p = hbarkand KE = p2/ 2m VM Ayres, ECE802-604, F13

  29. Reduced Dimensionality Systems: 2-DEG KE: write p in terms of hbark z px2 + py2 + nz2 ħ2p2 2m* 2m* 2m*Lz2 E = y Free motion in x and y directions Shown: Infinite potential well in z direction pz is constrained to be a number(s) x Graphene Thin Films Thin layers J.S. Moodera, Francis Bitter Magnet Lab, MIT A.K. Geim and K.S. Novoselov, Nat. Mater., 2007, 6, 183 VM Ayres, ECE802-604, F13

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

  31. Combine with this idea: e1 kx y-axis is E. The bowl is the KE that an e- has above the minimum requirement of ES required to be out of a bond p = hbarkand KE = p2/ 2m VM Ayres, ECE802-604, F13

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

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

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

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

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

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

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

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

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

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

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

  43. VM Ayres, ECE802-604, F13

  44. Use N(E) to get concentration nS VM Ayres, ECE802-604, F13

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

  46. VM Ayres, ECE802-604, F13

  47. Fermi wavenumber kf: VM Ayres, ECE802-604, F13

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

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

  50. 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 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 VM Ayres, ECE802-604, F13

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