Lecture VIITunneling

Tunneling

An electron of such an energy will never appear here!

classically

Ekin= 1 eV

0 V -2 V x

Potential barriers and tunneling

According to Newtonian mechanics, if the total energy is E, a particle that is on the left side of the barrier can go no farther than x=0. If the total energy is greater than U0, the particle can pass the barrier.

Tunneling – quantum approach

Schroedinger eq. for region x>L

EUdx

dm 02

22

2

)(2

022

2

EUm

dx

d

Solution: xAex )(

Potential barriers and tunneling

)(2

)(2

022

022 EU

mAeEU

meA xx

Two solutions: )(2

021 EUm

or )(2

022 EUm

Normalization condition: 1)(0

dxx

Solution: xAex 2)(

The probability to find a particle in the region II within

xxEUm

Axpr

002

20 )(

22exp)(

x

Potential barriers and tunneling

xxEUm

Axpr

002

20 )(

22exp)(

xAex 2)(

Potential barriers and tunneling

example

Let electrons of kinetic energy E=2 eV hit the barrier height of energy U0= 5 eV and the width of L=1.0 nm. Find the percent of electrons passing through the barrier?

LEU

mUE

UE

II

Tpad

trans )(2

2exp116 000 T=7.1·10-8

insulator

semiconductor

metalA

If L=0.5 nm.then T=5.2 ·10-4!

Scanning tunneling electron miscroscope

LeI 2

)(2

0 EUm

gdzie

Scanning tunneling electron miscroscope

Scanning tunneling electron miscroscope

Scanning tunneling electron miscroscope

Image downloaded from IBM, Almaden, Calif.It shows 48 Fe atoms arranged on a Cu (111) surface

Scanning tunneling electron miscroscope

particle decay

Approximate potential - energy function for an particle in a nucleus.

Tunneling

Nuclear fusion ( synteza ) is another example of tunneling effect

E.g. The proton – proton cycle