Lecture 4. Application to the Real World Particle 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

Wilson Ho (UC Irvine)

PIB Model for -Network in Conjugated Molecules LUMO  375 nm HOMO

(Engel, C5.1) 1,3,5-hexatriene LUMO 375 nm HOMO

Calculation done by Yoobin Koh

Origin of Color: -carotene

Origin of Color: chlorophyll

Solar Spectrum (Irradiation vs. Photon Flux) Maximum photon flux of the solar spectrum @ ~ 685 nm

Solar Spectrum (Irradiation vs. Photon Flux)

Understanding & Mimicking Mother Nature for Clean & Sustainable Energy: Artificial Photosynthesis

Organic Materials for Solar Energy Harvesting Bulk heterojunction solar cell with a tandem-cell architecture Hou, et al., (2008) Macromolecules Hole Electron Charge Separation Donor polymer acceptor

Essentially no thermal excitation Boltzmann Distribution (Engel, Section 2.1, P.5.2)

Particle in a finite height box: a potential well V(x) II III  is not required to be 0 outside the box.

Particle in a finite height box (bound states: E < V0) (1) the Schrödinger equation I II III  is not required to be 0 outside the box. (2) Plausible wave functions (3) Boundary conditions  -a/2 a/2

Particle in a finite height box: boundary condition I II III  (Engel, P5.7)

Particle in a finite height box: boundary condition II (4) the final solutions Region III Region I Region II -a/2 a/2 I II III 1.07 0.713 0.409 0.184 0.0461 (Engel, P5.7)

Particle in a finite height box: the final solutions tan (Example) V0 = 1.20 x 10-18 J, a = 1.00 nm E2 0.184 E4 0.713 E1 0.0461 E3 0.409 E5 1.07

Tunneling to classically-forbidden region

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valence electrons core electrons

From two to infinite array of Na atoms

Tunneling through a finite potential barrier (or U) (or L)

P5.1 8

Tunneling through a finite potential barrier Inside the barrier Outside the barrier Define alpha and represent equation Define k and represent equation

Tunneling through a finite potential barrier Inside the barrier Outside the barrier Ⅰ Ⅱ Ⅲ Assume that electrons are moving left to right. Boundary conditions Ⅰ / Ⅱ Ⅱ Ⅰ Ⅱ / Ⅲ Ⅲ

Transmission coefficient 4 equations for 4 unknowns. Solve for T. barrier width (decay length)-1

Probability current density (Flux)

Scanning Tunneling Microscope (STM) applications 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.

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

Quantum dot / Quantum well

How to put an elephant in a fridge? QM version no. 2

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).