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Understanding Two-Dimensional Electron Gas (2-DEG) and Mobility in Nanoelectronics

This lecture explores the concept of Two-Dimensional Electron Gas (2-DEG) within nanoelectronics, emphasizing the relationship between carrier concentration (nS) and electron mobility. As the density of states changes, it impacts mobility positively, illustrating how effective scattering decreases with higher electron concentrations. The discussion includes the implications of electronic confinement, band bending, and the significance of the Schrödinger equation in modeling electronic behavior. Practical examples of mobility measures and applications in semiconductor structures are also presented.

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Understanding Two-Dimensional Electron Gas (2-DEG) and Mobility in 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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