EE 60556: Fundamentals of Semiconductors Lecture Note

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EE 60556 Fundamentals of SemiconductorsLecture
Note 6 (09/22/09)Review of quantum mechanics

• Outline
• Last class Diffraction mapping the k-space,
  structure factor and atomic form factor (let us
  quickly review lecture 5 updated slides)
• Particle-wave duality and Schrodinger equation
• Particle-in-a-box (infinite and finite potential
  well)
• Tunneling
• Quantum harmonic oscillator

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Wave mechanics (single particle)
Time-independent Schrodinger equation
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Simple QM problems and solutions

• Simple free particle plane wave

QM free particle is like its classical
analog, Taking a continuum of energies.
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Simple QM problems and solutions

• Particle in 1-D box

Confinement leads to quantization.
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Simple QM problems and solutions

• Particle in 1-D box

Confinement leads to quantization. n quantum
number
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Simple QM problems and solutions

• Particle in finite potential well

Confinement leads to quantization. n quantum
number

• 30 sec discussions
• What are general solutions in each region?
• What are the boundary conditions?

We will revisit this problem in Kronig-Penney
model
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Simple QM problems and solutions

• Particle in finite potential well

Black lines RHS (right hand side) function Green
lines LHS and a0ap/4 gt 1 solution Pink lines
LHS and a0a2p gt 3 solutions
tan(2p)

• Symmetric well
• at least one solution
• wider the well ? more solutions
• Non-symmetric well
• - Might not have any solution

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Tunneling strictly a QM phenomenon
According to classical physics, a particle of
energy E less than the height U0 of a barrier
could not penetrate - the region inside the
barrier is classically forbidden. But the
wavefunction associated with a free particle must
be continuous at the barrier and will show an
exponential decay inside the barrier. The
wavefunction must also be continuous on the far
side of the barrier, so there is a finite
probability that the particle will tunnel through
the barrier. As a particle approaches the
barrier, it is described by a free particle
wavefunction. When it reaches the barrier, it
must satisfy the Schrodinger equation in the
form, which has the solution
http//hyperphysics.phy-astr.gsu.edu/hbase/quantum
/barr.html
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Quantum harmonic oscillators (Kittel Appendix C
for detailed derivation)

• Quantum harmonic oscillator describes photons,
  phonons, vibration of molecules etc.
• Quantum harmonic oscillator is like its classic
  counterpart an object on a spring, described by
  a characteristic frequency (that is why we call
  it harmonic) and its potential energy depends on
  the square of the displacement from equilibrium.
• The unique features of a quantum harmonic
  oscillator are
• Its energy states (eigen states) are quantized
• Its ground state energy is NOT ZERO
• The energy difference between adjacent states is
• It is characteristic frequency is determined by
  the force constant and the vibrating object
  mass. For a diatomic molecule shown on the left,
  a reduced mass mr should be used.

http//hyperphysics.phy-astr.gsu.edu/Hbase/quantum
/hosc.htmlc1