ECE 606 Lecture 4: Solution of Schrodinger Equation

By Muhammad A. Alam

Purdue University

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Abstract

Outline:
  • Time-independent Schrodinger Equation
  • Analytical solution of toy problems
  • Bound vs. tunneling states
  • Conclusions
  • Additional Notes: Numerical solution of Schrodinger Equation

References

R. F. Pierret, "Advanced Semiconductor Fundamentals", Modular Series on Solid State Devices, Volume VI, Addison-Wesley, 1987, Chapter 2 (pages 29-45)

Cite this work

Researchers should cite this work as follows:

  • Muhammad A. Alam (2009), "ECE 606 Lecture 4: Solution of Schrodinger Equation," https://nanohub.org/resources/5756.

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Time

Location

EE 115, Purdue University, West Lafayette, IN

Tags

  1. course lecture
  2. device physics
  3. devices
  4. quantum mechanics
  5. transistors
ECE 606 Lecture 4: Solution of Schrodinger Equation
  • EE-606: Solid State Devices Lecture 4: Solution of Schrodinger Equation 1. EE-606: Solid State Devices Le… 0
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  • Outline 2. Outline 14.533333333333333
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  • Motivation 3. Motivation 50.966666666666669
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  • Time-independent Schrodinger Equation 4. Time-independent Schrodinger E… 220.6
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  • Time-independent Schrodinger Equation 5. Time-independent Schrodinger E… 464.63333333333333
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  • A Simple Differential Equation 6. A Simple Differential Equation 617.9666666666667
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  • Outline 7. Outline 729.7
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  • Full Problem Difficult: Toy Problems First 8. Full Problem Difficult: Toy Pr… 731
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  • Five Steps for Analytical Solution 9. Five Steps for Analytical Solu… 754.76666666666665
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  • Case 1: Bound-levels in Finite Well (steps 1,2) 10. Case 1: Bound-levels in Finite… 806.5333333333333
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  • Step3: Continuity of wave-function 11. Step3: Continuity of wave-func… 1059.5333333333333
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  • Step 3: Continuity of Wavefunction 12. Step 3: Continuity of Wavefunc… 1225.9
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  • Step 4: Bound-level in Finite Well 13. Step 4: Bound-level in Finite … 1287.1
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  • Step 4: Graphical Method for Bound Levels 14. Step 4: Graphical Method for B… 1427.3666666666666
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  • Step 4: Graphical Method for Bound Levels 15. Step 4: Graphical Method for B… 1586.3666666666666
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  • Step 5: Wave-functions 16. Step 5: Wave-functions 1638.8666666666666
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  • Step 5: Calculating Wave-function 17. Step 5: Calculating Wave-funct… 1793.1333333333334
    00:00/00:00
  • Aside: Infinite Quantum Well 18. Aside: Infinite Quantum Well 1859.8666666666666
    00:00/00:00
  • Five steps for Analytical Solution: Follow rules 19. Five steps for Analytical Solu… 2032.4333333333334
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  • Practical examples: Si/Ag and Si/SiO2 20. Practical examples: Si/Ag and … 2055.6333333333332
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  • Outline 21. Outline 2221.3666666666668
    00:00/00:00
  • Case 2: Solution for Particles with E>>U 22. Case 2: Solution for Particles… 2230.2333333333331
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  • Free Particle … 23. Free Particle … 2344.3333333333335
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  • Case 3: Bound vs. Tunneling State 24. Case 3: Bound vs. Tunneling St… 2405.3333333333335
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  • Practical Example: Oxide Tunneling 25. Practical Example: Oxide Tunne… 2613.8666666666668
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  • Conclusions 26. Conclusions 2731.7666666666669
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  • Outline 27. Outline 2871.3333333333335
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  • Numerical solution of Schrodinger Equation 28. Numerical solution of Schrodin… 2926.0333333333333
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  • (1) Define a grid … 29. (1) Define a grid … 2976
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  • Aside: Finite difference – Connecting neighbors 30. Aside: Finite difference – C… 3080.9333333333334
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  • (2) Express equation in Finite Difference Form 31. (2) Express equation in Finite… 3184.7
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  • (3) Define the matrix … 32. (3) Define the matrix … 3292.6
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  • (4) Solve the Eigen-value Problem 33. (4) Solve the Eigen-value Prob… 3386.9
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