Lecture physics a2 barrier penetration and tunneling huynh quang linh

 “All of modern physics is governed by that magnificent and thoroughly confusing discipline called quantum mechanics It has survived all tests and there is no reason to believe that there is any flaw[.]

 “All of modern physics is governed by that

magnificent and thoroughly confusing discipline

called quantum mechanics It has survived all tests and there is no reason to believe that there is any flaw in it….We all know how to use it and how to

apply it to problems; and so we have learned to live with the fact that nobody can understand it.”

Murray Gell-Mann

Lecture 9: Barrier

Penetration and Tunneling

x

0 L

U0

x

U(x)

E

U(x)

nucleus

 How quantum particles tunnel

 Nuclear Decay

 Solar Fusion

NH Maser

 Due to “barrier penetration”, the electron density of a metal actually extends outside the surface of the metal!

E F

Occupied levels

Work function F

V o

Assume that the work function (i.e., the energy difference between the most energetic conduction electrons and the potential barrier

at the surface) of a certain metal is F = 5 eV Estimate the

distance x outside the surface of the metal at which the electron probability density drops to 1/1000 of that just inside the metal.

1000

1 )

0 (

)

2

2



  Kx

e

x





x = 0

x

nm K

1000

1 ln

2

1  















“Leaky” Particles: Revisited

0

1.505

h eV nm



using

Application: Tunneling Microscopy

 Due to the quantum effect of “barrier

penetration,” the electron density of a

material extends beyond its surface:

material STM tip

STM tip material

~ 1 nm

x Metal

tip

One can exploit this

to measure the

electron density on a

material’s surface:

Na atoms

on metal:

Real STM tip

STM images

DNA Double Helix:

www-aix.gsi.de/~bio

x

U=0

U=U o

 What is the “Transmission

Coefficient T”, the probability

an incident particle tunnels

through the barrier?

 Consider a barrier (II) in the

middle of a very wide infinite

square well.

Tunneling Through a Barrier (1)

Region I:I ( x )  A1 sin kx  A2 cos kx E > U: oscillatory solution

Region III:III ( x )  C1 sin kx  C2 cos kxE > U: oscillatory solution

Kx Kx

II ( x )  B1e  B2e



 Next we would need to apply the “continuity conditions” for both

 and d/dx at the boundaries x = 0 and x = L to determine the

A, B, and C coefficients

 To get an “exact” result describing how quantum particles penetrate this barrier, we write the proper wavefunction in each of the three regions shown in Figure:

Tunneling Through a Barrier (2)

This is nearly the same result as in the “leaky particle” example!

Except for G:

• G slightly modifies the

transmission probability

• G arises from the fact

that the amplitude at

x = 0 is not a maximum

 In general the tunneling coefficient T

can be quite complicated (due to the

contribution of amplitudes “reflected”

off the far side of the barrier). 0 L

U 0

x

U(x)

E

 However, in many situations, the barrier width L is much

larger than the ‘decay length’ 1/K of the penetrating wave;

in this case ( KL >> 1 ) the tunneling coefficient simplifies to:

2KL



where

16 E 1 E

G

U U

0 1 2 3 4

Tunneling Through a Barrier (3)

KL

e

 The plot illustrates how the

transmission coefficient T changes

as a function of barrier width L ,

for two different values of the

particle energy.

*In fact, some references (wrongly) completely omit G (including Phys 214 before 2006!) We will state when you can ignore G

0 L

U 0

x

U(x)

E

2KL

m

K  2 2 0 



where

16 E 1 E

G

U U

0 0.2 0.4 0.6

0.5 0.75 1 1.25 1.5

L

T E=2/3 U 0

E=1/3 U 0

 T depends on the energy below the barrier (U0-E) and on

the barrier width L.

 By far the dominant effect is the decaying exponential*:

0 L

U 0

x

U(x)

E

Example: Barrier Tunneling in an STM

 Let’s consider a simple problem:

An electron with a total energy of E=6 eV

approaches a potential barrier with a

height of U o = 12 eV If the width of the

barrier is L=0.18 nm, what is the

probability that the electron will tunnel

metal STM tip

air gap

0 L

U 0

x

U(x)

E

2KL

2(12.6)(0.18)

metal STM tip

air gap

1.505 eV-nm

h

Question: What will T be if we double the width of the gap?

Example: Barrier Tunneling in an STM

 Let’s consider a simple problem:

An electron with a total energy of E=6 eV

approaches a potential barrier with a

height of U o = 12 eV If the width of the

barrier is L=0.18 nm, what is the

probability that the electron will tunnel

through the barrier?

1 1

16 1 16 1 4

2 2

E E G

U U