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

Wilson Ho (UC Irvine)

HOMO

LUMO

375 nm

PIB Model for -Network in Conjugated Molecules

HOMO

LUMO

375 nm

(Engel, C5.1)

1,3,5-hexatriene

Calculation done by Yoobin Koh

Origin of Color: -carotene

Origin of Color: chlorophyll

Maximum photon flux of the solar spectrum @ ~ 685 nm

Solar Spectrum (Irradiation vs. Photon Flux)

Solar Spectrum (Irradiation vs. Photon Flux)

Understanding & Mimicking Mother Nature for Clean & Sustainable Energy:

Artificial Photosynthesis

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

Boltzmann Distribution(Engel, Section 2.1, P.5.2)

Essentially no thermal excitation

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 I

I II III

(Engel, P5.7)

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

Tunneling to classically-forbidden region

.

coreelectrons

valenceelectrons

From two to infinite array of Na atoms

Tunneling through a finite potential barrier

(or L)

(or U)

P5.18

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

4 equations for 4 unknowns. Solve for T.

Transmission coefficient

barrier width

(decay length)-1

Probability current density (Flux)

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

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).