Lecture 4. Application to the Real World Particle in a

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Lecture 4. Application to the Real
WorldParticle in a Finite Box (Potential
Well)Tunneling through a Finite Potential Barrier
References

• Engel, Ch. 5
• Molecular Quantum Mechanics, Atkins Friedman
  (4th ed. 2005), Ch. 2.9
• Introductory Quantum Mechanics, R. L. Liboff
  (4th ed, 2004), Ch. 7-8
• A Brief Review of Elementary Quantum Chemistry
• http//vergil.chemistry.gatech.edu/notes/quantrev
  /quantrev.html
• Wikipedia (http//en.wikipedia.org) Search for
• Finite potential well
• Finite potential barrier
• Quantum tunneling
• Scanning tunneling microscope

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Wilson Ho (UC Irvine)
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PIB Model for ?-Network in Conjugated Molecules
LUMO
375 nm
HOMO
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(Engel, C5.1) 1,3,5-hexatriene
LUMO
375 nm
HOMO
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Calculation done by Yoobin Koh
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Origin of Color ?-carotene
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Origin of Color chlorophyll
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Solar Spectrum (Irradiation vs. Photon Flux)
Maximum photon flux of the solar spectrum _at_ 685
nm
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Solar Spectrum (Irradiation vs. Photon Flux)
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Understanding Mimicking Mother Nature for
Clean Sustainable Energy Artificial
Photosynthesis
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Organic Materials for Solar Energy Harvesting
Bulk heterojunction solar cell with a
tandem-cell architecture
Hou, et al., (2008) Macromolecules
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Essentially no thermal excitation
Boltzmann Distribution (Engel, Section 2.1, P.5.2)
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Particle in a finite height box a potential well
V(x)
I
II
III
? is not required to be 0 outside the box.
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Particle in a finite height box (bound states E
lt V0)
(1) the Schrödinger equation
I
II
III
? is not required to be 0 outside the box.
(2) Plausible wave functions
(3) Boundary conditions
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Particle in a finite height box boundary
condition I
I
II
III
(Engel, P5.7)
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Particle in a finite height box boundary
condition II
(4) the final solutions
1.07
0.713
0.409
0.184
0.0461
(Engel, P5.7)
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Particle in a finite height box the final
solutions
(Example) V0 1.20 x 10-18 J, a 1.00 nm
E4 0.713
E2 0.184
E3 0.409
E1 0.0461
E5 1.07
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Tunneling to classically-forbidden region
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valence electrons
core electrons
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From two to infinite array of Na atoms
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Tunneling through a finite potential barrier
(or U)
(or L)
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P5.1
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Tunneling through a finite potential barrier
Inside the barrier
Outside the barrier
Define alpha and represent equation
Define k and represent equation
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Tunneling through a finite potential barrier
Inside the barrier
Outside the barrier
?
?
?
Assume that electrons are moving left to right.
Boundary conditions
? / ?
?
?
? / ?
?
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Transmission coefficient
4 equations for 4 unknowns. Solve for T.
barrier width
(decay length)-1
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Probability current density (Flux)
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Scanning Tunneling Microscope (STM)
applications

• Introduced by G. Binnig and W. Rohrer at the IBM
  Research Laboratory in 1982 
• (Noble Prize in 1986)

Basic idea

• Electron tunneling current depends on the barrier
  width and decay length.
• STM measures the tunneling current to know the
  materials depth and surface profiles.

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Modes of Operation

• Constant Current Mode
• Tips are vertically adjusted along the constant
  current 

• Constant Height Mode
• Fix the vertical position of the tip

• Barrier Height Imaging
• Inhomogeneous compound

• Scanning Tunneling Spectroscopy
• Extension of STM this mode measured the density
  of electrons in a sample

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Quantum dot / Quantum well
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How to put an elephant in a fridge? QM version
no. 2
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How to put an elephant in a fridge? QM version
no. 2
??? ?? ???. ???? ???? ?? ????. ? ??? ???? ?????
??? ?? ???? ???? ???? ????.
Close the fridge door. Make the elephant run to
the fridge. Repeating this for infinite times,
the elephant will eventually enter the fridge
through the door (by quantum tunneling).