## ECE 606 Lecture 3: Emergence of Bandstructure

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**Table of Contents:**

00:00 ECE606: Solid State Devices Lecture 3

00:24 Motivation

01:17 Time-independent Schrodinger Equation

02:22 Time-independent Schrodinger Equation

04:23 A Simple Differential Equation

05:29 Presentation Outline

05:46 Full Problem Difficult: Toy Problems First

06:07 Case 1: Solution for Particles with E>>U

06:57 Free Particle …

07:57 Full Problem Difficult: Toy Problems First

08:02 Case 2: Bound State Problems

10:53 1-D Particle in a Box – A Solution Guess

11:57 1-D Particle in a Box – Visualization

13:01 1-D Particle in a Box – Normalization to ONE particle

15:02 1-D Particle in a Box – The Solution

16:06 1-D Particle in a Box – Quantum vs. Macroscopic

17:17 Presentation Outline

17:26 Full Problem Difficult: Toy Problems First

17:33 Five Steps for Closed System Analytical Solution

19:03 Case 2: Bound-levels in Finite Well (steps 1,2)

20:36 Case 2: Bound-levels in Finite Well (steps3)

23:39 Case 2: Bound-levels in Finite Well (step 4)

25:08 Case 2: Bound-levels in Finite Well (steps 4 graphical)

26:18 Case 2: Bound-levels in Finite Well (steps 4 graphical)

26:49 Case 2: Bound-levels in Finite Well (steps 5 wavefunction)

27:12 Step 5: Wave-functions

27:42 Key Summary of a Finite Quantum Well

29:16 Presentation Outline

29:24 Transmission through a single barrier Scattering Matrix approach

30:17 Tunneling through a single barrier

30:48 Single barrier case

32:38 Generalization to Transfer Matrix Method

33:31 Single barrier case

36:38 Single barrier : Concepts

39:25 Effect of barrier thickness below the barrier

40:29 Single Barrier – Key Summary

41:00 Presentation Outline

41:05 Double Barrier Transmission: Scattering Matrix approach

41:17 Reminder: Single barrier

41:52 Double barrier: Concepts

42:37 Double barrier: Quasi-bound states

43:23 Effect of barrier height

44:59 Effect of barrier thickness

45:00 Double barrier energy levels Vs Closed system

45:31 Particle in a box

45:44 Double barrier & particle in a box

46:33 Open systems Vs closed systems

47:36 Reason for deviation?

48:30 Double Barrier Structures - Key Summary

49:07 Presentation Outline

49:14 1 Well => 1 Transmission Peak

49:37 2 Wells => 2 Transmission Peaks

50:09 3 Wells => 3 Transmission Peaks

50:19 4 Wells => 4 Transmission Peaks

50:20 5 Wells => 5 Transmission Peaks

50:21 6 Wells => 6 Transmission Peaks

50:24 7 Wells => 7 Transmission Peaks

50:25 8 Wells => 8 Transmission Peaks

50:39 9 Wells => 9 Transmission Peaks

50:40 19 Wells => 19 Transmission Peaks

50:42 29 Wells => 29 Transmission Peaks

50:44 39 Wells => 39 Transmission Peaks

50:45 49 Wells => 49 Transmission Peaks

51:20 N Wells => N Transmission Peaks

52:30 1 Well => 1 Transmission Peak => 1 State

52:37 2 Wells => 2 Transmission Peaks => 2 States

52:39 3 Wells => 3 Transmission Peaks => 3 States

52:39 4 Wells => 4 Transmission Peaks => 4 States

52:40 5 Wells => 5 Transmission Peaks => 5 States

52:40 6 Wells => 6 Transmission Peaks => 6 States

53:05 7 Wells => 7 Transmission Peaks => 7 States

53:05 8 Wells => 8 Transmission Peaks => 8 States

53:06 9 Wells => 9 Transmission Peaks => 9 States

53:06 19 Wells => 19 Transmission Peaks => 19 States

53:09 29 Wells => 29 Transmission Peaks => 29 States

53:09 39 Wells => 39 Transmission Peaks => 39 States

53:11 49 Wells => 49 Transmission Peaks => 49 States

54:05 N Wells => N Transmission Peaks => N States

54:53 N Wells => N States => 1 Band

55:07 N Wells => 2N States => 2 Bands

55:48 N Wells => 2N States => 2 Bands

55:59 X States/Well => X Bands

57:10 X States/Well => X Bands

57:37 Formation of energy bands

57:59 Presentation Outline

58:41 Five Steps for Closed System Analytical Solution

59:10 Open System: Generalization to Transfer Matrix Method

59:36 Presentation Outline

59:39 Numerical solution of Schrodinger Equation

60:09 (1) Define a grid …

60:33 Second Derivative on a Finite Mesh

62:06 (2) Express equation in Finite Difference Form

62:56 (3) Define the matrix …

63:44 (4) Solve the Eigen-value Problem

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CIVL 2104, Purdue University, West Lafayette, IN