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ECE 874: Physical Electronics

ECE 874: Physical Electronics. Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University ayresv@msu.edu. Lecture 10, 02 Oct 12. Answers I can find:. Working tools:. Connection: conservation of energy and working tool 2: the Schroedinger equation.

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ECE 874: Physical Electronics

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  1. ECE 874:Physical Electronics Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University ayresv@msu.edu

  2. Lecture 10, 02 Oct 12 VM Ayres, ECE874, F12

  3. Answers I can find: VM Ayres, ECE874, F12

  4. Working tools: VM Ayres, ECE874, F12

  5. Connection: conservation of energy and working tool 2: the Schroedinger equation VM Ayres, ECE874, F12

  6. VM Ayres, ECE874, F12

  7. VM Ayres, ECE874, F12

  8. VM Ayres, ECE874, F12

  9. VM Ayres, ECE874, F12

  10. Two unknowns y(x) and E in eV from one equation? 1. You can find y(x) by inspection whenever the Schroedinger equation takes a form with a known solution like and exponential. The standard form equation will also give you one relationship for kx. 2. Matching y(x) at a boundary puts a different condition on kx and setting kx = kx enables you to also solve for E in eV. VM Ayres, ECE874, F12

  11. Or equivalent Aexpikx + Bexp-ikx form Infinite potential well VM Ayres, ECE874, F12

  12. With B = 0: tunnelling out of a finite well VM Ayres, ECE874, F12

  13. Finite Potential Well: (eV) Electron energy: E > U0 Electron energy: E < U0 (nm) Regions: -∞ to 0 0 to a a to +∞ VM Ayres, ECE874, F12

  14. Finite Potential Well: (eV) Electron energy: E < U0 (nm) Region: 0 to a VM Ayres, ECE874, F12

  15. Finite Potential Well VM Ayres, ECE874, F12

  16. Finite Potential Well: (eV) Electron energy: E < U0 (nm) Regions: -∞ to 0 a to +∞ VM Ayres, ECE874, F12

  17. Finite Potential Well VM Ayres, ECE874, F12

  18. Finite Potential Well New: e- goes away at ±∞ New boundary matching condition New: e- exists outside of well region VM Ayres, ECE874, F12

  19. Finite Potential Well Gives a decreasing exponential e-a|x| in this region VM Ayres, ECE874, F12

  20. Finite Potential Well VM Ayres, ECE874, F12

  21. Finite Potential Well y-(x) and y0(x) are done to within A0. If you need A0, use Working Tool 3: the existence theorem, in the easy region: 0 < x < a. VM Ayres, ECE874, F12

  22. Finite Potential Well To find B+ in terms of A0 to complete y+(x) add 2.41b and 2.41d, and re-arrange to get B+: (2.41a) (2.41b) (2.41c) (2.41d) VM Ayres, ECE874, F12

  23. Finite Potential Well y-(x), y0(x) and y+(x) are done to within A0. If you need A0, use Working Tool 3: the existence theorem, in the easy region: 0 < x < a. Wave functions that represent e- are found. Now find its total energy E in eV. VM Ayres, ECE874, F12

  24. Finite Potential Well .42) VM Ayres, ECE874, F12

  25. Finite Potential Well .42) This is basically the solution for E in eV. VM Ayres, ECE874, F12

  26. Finite Potential Well VM Ayres, ECE874, F12

  27. Finite Potential Well Red: LHS curve Blue: RHS curve Solve graphically: LHS = tan(…E) RHS = polynomial (…E) Where they intersect is the value for E in eV E in eV VM Ayres, ECE874, F12

  28. Finite Potential Well Red: LHS curve Blue: RHS curve Solve graphically: LHS = tan(…E) RHS = polynomial (…E) Where they intersect is the value for E in eV Quantized E1, E2, E3,.. for the finite well too, since tan(…E) repeats itself in multiples of p/2 E in eV VM Ayres, ECE874, F12

  29. En in eV Finite Potential Well: These are the energies En for the e- in the well, but the values are consistent with the physical situation that the well has a finite height U0 and that the e- can tunnel into the out of well regions on either side. (eV) Electron energy: E < U0 (nm) Regions: -∞ to 0 0 to a a to +∞ VM Ayres, ECE874, F12

  30. Finite Potential Well Advantage is: you scale to well height U0 and width a. Note that width a only affects the LHS: the number/spacing of tan curves. VM Ayres, ECE874, F12

  31. Finite Potential Well Red: LHS curve Blue: RHS curve Solve graphically: LHS = tan(…E/U0) RHS = polynomial (…E/U0) Where they intersect are the values for En/U0 in eV VM Ayres, ECE874, F12

  32. (a) Shallow well U0, single intersection for E1 (b) Deeper well U0, more intersections for E1, E2, E3,…. (c) Comparison of finite (solid) and infinite (dotted) well energy levels En shows that the infinite well solution progressively over-estimates the higher En VM Ayres, ECE874, F12

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