Today: Examples of Tunneling - · PDF fileform and has lots of KE. How does this happen

Today: Examples of Tunneling

• HWK13 will be available, due Wed. after the break.• Reading for break: Chapter 8 is good, but look at

anything else in the book that’s interesting. You should be able to understand it now.

1. Scanning tunneling microscope.

2. Alpha particle tunneling

Use tunneling to measure small changes in distance. Nobel prize winning idea- invention of “scanning tunnelingmicroscope (STM)”. Measure atoms on surfaces.

White dots are individual atoms!

Black dots are missing atoms!

Very important nanotech tool. Can’t see… can’t study.

Picture of STM-- voltage applied between tip and sample. Holds potential difference constant so electron current flows into the tip. Figure out what potential energy looks like in different regionsso can calculate current, determine sensitivity to gap distance.

energy

Sampletip

Tip

I

SA

MP

LE M

ETA

L

Tip

I

SA

MP

LE M

ETA

L

+

What does electron potentialenergy in the tip look like?

a. higher than V sampleb. same as V samplec. lower than V sampled. tilts downward from left to righte. tilts upward from left to right

applied voltage

Picture of STM-- voltage applied between tip and sample. Potential energy in different regions so can calculate current, determine sensitivity to gap distance.

energy

Sampletip

Tip

I

SA

MP

LE M

ETA

L

Tip V

I

SA

MP

LE M

ETA

L

+

applied voltage

What is the electron potential energy in the air gap?

linear connection

Notice changing V will change barrier, and hencetunneling current.

Tip

V

I

SAMPLE METAL +

Tip

V

I

SAMPLE METAL +

cq. if tip is moved closer to sample which picture is correct?

a. b. c. d.

tunneling current will go: (a) up, (b) stay same, (c) go down(a) go up. a is smaller, so e-2αa is bigger (not as small), T bigger

STM (picture with reversed voltage, works exactly the same)

end of tip always atomically sharp

How sensitive to distance?Need to look at numbers.

T ~e-2αa

how big is α?(= 1/(how long is exponentialtail of wave function =how far can tunnel).

α = E)2m(V0 −

Calculate: if V0-E = 4 eV, α = 1/(1 x 10-10 m)

So if a is 3 x 10-10 m, T = e-6 = .0025add 1 extra atom (d ~ 10-10 m),how much does T change?

T = e-4 =.018 so one atom more 7x the current!

actual STM moves tip across surface, adjusts distanceto keep distance constant, keeps track of how much hasto move in and out to make map.

Radioactive alpha decayNucleus is unstable → emits a particle Typically found for large atoms with lots of protons and neutrons.

Radon-22286 protons, 136 neutrons

Proton (positive charge)Neutron (no charge)

Nucleus has lots of protons and lots of neutrons. Two forces acting in nucleus: - Coulomb force .. Protons really close together, so very big repulsion from coulomb force- Nuclear force (attraction between nuclear particles is very strong if very close together) … called the STRONG Force.

Alpha Decay: Nucleus emits an alpha particleAn alpha particle is 2 neutrons and 2 protons.

Radioactive decay

Radon-22286 protons, 136 neutrons

Proton (positive charge)Neutron (no charge)

In alpha-decay, an alpha-particle is emitted from the nucleus.

This raises the ratio of neutrons to protons … makes for a more stable atom. (Neutron are neutral.. no coulomb repulsion, but nuclear force attraction)

How does this happen…Starting point always to look at potential energy curve for particle

Nucleus(Z protons,

Bunch o’ neutrons)

New nucleus(Z-2 protons,

Bunch o’ neutrons)

+Alpha particle(2 protons, 2 neutrons)

Look at this system… as the distance between the alpha particle and the nucleus changes.

V=0 At a great distance

(Z-2)

As bring α closer, What happens to potential energy?



As bring α closer, what happens to potential energy?

V(r)

V(r)V(r)A

B

C

D. Something else

r

r

r



As bring α closer, What happens to potential energy?

V(r)B

Takes energy to push α towards the nucleus, so potential energy must increase.

reeZk

rqkqrV )2)()(2()( 21 −

==


V(r)

Edge of the nucleus (~8x10-15 m), Nuclear (Strong) force starts actingStrong attraction between nucleonsPotential energy drops dramatically

30 MeV

4 to 9MeVof KE

Starting point always to look at potential energy curve for particle

V(r)

Edge of the nucleus (~8x10-15 m), Nuclear (Strong) force starts actingStrong attraction between nucleonsPotential energy drops dramatically

30 MeV

r

Ene

rgy

Bring alpha-particle closer

Coulomb force dominates

Coulomb&Nuclear

reeZk

rqkqrV )2)()(2()( 21 −

==

Nuclear Physics Sim

Observations show Alpha-particles exit with a range of energies

V(r)30 MeV

4 to 9MeVof KE

Inside αparticle can form and has lots of KE


V(r)30 MeV

100MeV of KE

4MeV of KE

Observe α-particles from different isotopes (same protons, different neutrons), exit with different amounts of energy.

V(r)

30 MeV (Same peak height if protons same)

4MeV KE9MeV KE

Was one or the other more likely to tunnel through? And WHY??? a. the 9MeV alpha particle was more likelyb. the 4MeV alpha particle was more likely c. both had the same probability of tunneling

)(22 EVm

−=αDecay constant:

Observe a particles from different isotopes (same protons, different neutrons), exit with different amounts of energy.

V(r)30 MeV

4MeV KE9MeV KE

Answer is a. The 9 MeV electron more probable…

1. Less distance to tunnel, 2. Decay constant always smaller3. Wave function doesn’t decay as much before reaches other side … more probable!

Isotopes that emit higher energy alpha particles, have shorter lifetimes!!!

)(22 EVm

−=α

Solving Schrodinger equation for this potential energy is hard!

V(x) V(x)

Square barrier is much easier…and get almost the same answer!

Quantum Tunneling Sim

Rest of today:other applications of tunneling in real world

Scanning tunneling microscope (STM):how QM tunneling lets us map individual atoms on surface

Interesting example not time to cover but in notes:• Sparks and corona discharge (also known as field

emission) electrons popping out of materials when voltage applied.

• Many places including plasma displays.

A. stop.B. be reflected back.C. exit the wire and keep moving to the right.D. either be reflected or transmitted with some probability.E. dance around and sing, “I love quantum mechanics!”

If the total energy E of the electron is LESS than the work function of the metal, V0, when the electron reaches the end of the wire, it will…

warm up on whatelectron does at barrierthen apply

a more common manifestation of QM tunneling1. understanding discharges- electrons popping out ofsurface when voltage applied.

What electric field needed to rip electron out of solid if no tunneling?

+ -r

+ -r

+ -r

+ -r

+ -r

+ -r+ -

rV

solid

E

Applied E must exceed E of nucleus.Using H to estimate Enuc.E ~ kq/r2

~ 26 V/ 0.053 nm~ 5 x 1011 V/m~ 5 x 109 V/cm

so would need around 5 x 109 V/cm J. Travoltage sim

Get few billion volts from rubbing feet on rug?

NO! Electrons tunnel out at much lower voltage.

V

1 2 3

What is the minimum we need to know to figure out tunneling probability?

a. only db. only Vc. V and d d. V, d, and work functions of finger and doorknobe. none of the above, need additional information

d

ans. d. if have these, can get potentials, solve Schrod Eqn

so would need around 2.6 x 109 V/cm J. Travoltage sim

Get few billion volts from rubbing feet on rug?

NO! Electrons tunnel out at much lower voltage.

V

1 2 3d

Energy

x

U

E

V = 0, T ~e-2αa tiny.Rub feet, what happensto potential energy?

Distance to tunnel much smaller. Big V ⇒ a small, so e-2αa big enough, e’s tunnel out!

WorkFunction Of finger

WorkFunction Of doorknob

Potential difference between finger/door

d

energy

SA

MP

LE M

ETA

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Tip

SA

MP

LE M

ETA

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tip

x

Book description of STM wrong.

Says looks like this, and one looks at tunnelingcurrent from sample to tip to measure gap.

What is wrong with this?

Electron tunnels from sampleto tip. What would V(x) look like then?

a. same as before. b. V in tip higher, V sample lower.c. V in tip lower, V sample higher.d. V same on each side as before

but barrier higher.

-

ans. b. electron piled on top (in energy) of many other electronsthat contribute to V(x). Add electron, makes higher V(x),remove makes lower. So what does next electron want to do?

Documents

Today: Examples of Tunneling - · PDF fileform and has lots of KE. How does this happen