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 Chemistryhttp://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
Hou, et al., (2008) Macromolecules
Bulk heterojunction solar cell with a tandem-cell architecture
Organic Materials for Solar Energy Harvesting
Hole
Electron
Charge Separation
Donor polymer
acceptor
Particle in a finite height box: a potential well V(x)
I II III
is not required to be 0 outside the box.
Particle in a finite height box (bound states: E < V0)
I II III
(2) Plausible wave functions
(3) Boundary conditions
(1) the Schrödinger equation
a/2
a/2
is not required to be 0 outside the box.
Particle in a finite height box: boundary condition II
Region III
Region I
Region II
I II III
(4) the final solutions
a/2
a/2
(Engel, P5.7)
0.04610.184
0.409
0.713
1.07
02 2 2
2 2 2tan or tan
2 2
m V Ea mE mE ak k
02 2 2
2 2 2cotan or cot
2 2
m V Ea mE mE ak k
2 2 20
2 2 2cot2 2 2
m V E a mEa mEa
2 2 20
2 2 2cot2 2 2
m V E a mEa mEa
tan
Particle in a finite height box: the final solutions
(Example) V0 = 1.20 x 10-18 J, a = 1.00 nm
E1
0.0461E3
0.409
E2
0.184
E4
0.713
E5
1.07
Inside the barrier
Define alpha and represent equation
Outside the barrier
Define k and represent equation
Tunneling through a finite potential barrier
Assume that electrons are moving left to
right.
Boundary conditions
Ⅰ Ⅱ Ⅲ
Ⅰ
Ⅱ
Ⅲ
Ⅰ / Ⅱ
Ⅱ / Ⅲ
Tunneling through a finite potential barrier
Inside the barrier
Outside the barrier
Scanning Tunneling Microscope (STM)
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.
applications
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
How to put an elephant in a fridge? QM version no. 2
냉장고 문을 닫는다 .코끼리가 냉장고를 향해 돌진한다 .이 과정을 반복하면 양자터널링 현상에 의해 언젠가는 코끼리가 냉장고에 들어간다 .
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).