Quantum TunnellingQuantum Tunnelling

Quantum Physics2002

Recommended Reading:Recommended Reading:R.Harris, Chapter 5 Sections 1, 2 and 3

Potential Barrier: Potential Barrier: E < UE < U00

0kdx

d

1212

2

φ 0k

dx

d 2

12

2

3φ0

dx

d

22

2

2

φα

Region I Region IIIRegion II

202 EUm2

α

xL 0

Lx 0 U

0x 0

)x(U 0

Potential

221

mE2k

where

and

x

U = U0

E = K.E.

I II

x =0

U

III

x =L

1

E = K.E.

Region I: xikxik 11 BeAex 1φ 2

Wavefunctions Wavefunctions

Incident Reflected

Region II: xx DeCex αα2φ

3

Region III: xikxik 11 GeFex 3φ

Transmitted Left Moving No

term because there is no particle incident from the right.

xik1Ge

4

Must keep both terms.Do you see why?

Boundary ConditionsBoundary Conditions

00 21 φφ 000ik0ik DeCeBeAe 11 αα

Match wavefunction and derivative at x = 0.

D CBA 5

000ik1

0ik1 DeCeBeikAeik 11 αα αα

0

2

0

1dx

dφ

dx

dφ

DCBAik 1 α 6

LL 32 φφ LikLL 1FeDeCe αα 7

Lik1

LL 1FeikDeCe αα αα L

3

L

2dx

dφ

dx

dφ8

Match wavefunction and derivative at x = L.

Boundary ConditionsBoundary ConditionsWe now have 4 equations and 5 unknowns, Can solve for B, C, D and F in terms of A. This is left as an exercise, A lot of algebra but nothing complicated!!

Again we define a Reflection and Transmission coefficient:

2

2

*

*

A

B

AA

BB current inc current refl

R

2

2

*

*

1

3

A

F

AA

FFkk

current inc. current trans.

T Since k1 = k3.

Substituting for B and F in terms of A gives:

222

121

22

2

k k4Lsinh

LsinhR

ααα

αReflection Coefficient R 9

R and T Coefficients:R and T Coefficients:

We can write k1 and k2 in terms of E and U0.

00

02

00

UE

1UE

4LEUm2

sinh

UE

1UE

4T

222

122

21

2

2221

21

2

k kk4Lsinh

k k4T

αα

αα

Transmission Coefficient T

2

022

UEm2k

221

mE2k

This then gives

10

11

Recall that sin(i) =sinh().

00

02

02

UE

1UE

4LEUm2

sinh

LEUm2

sinh

R

similarly we can find an expression for the Reflection coefficient

11a

Dividing across by

00 UE

1UE

4 gives

1

0

220

EUE4L sinhU

1T

α

12

or rearranging

1-

220

0

LsinhU

EUE41R

α

12a

Graph of Transmission ProbabilityGraph of Transmission Probability

U0 = 0.1 eV

U0 = 1.0 eV

U0 = 5.0 eV

U0 = 10.0 eV

T

E/U0

Transmission curves for a barrier of constant width 1.0 nm with different heights U0

L = 1.0 nm

L = 0.5 nm

L = 0.1 nm

TE/U0

Transmission curves for a barrier of constant height 1.0 eV for a series of different widths L.

100

10-5

10-10

10-15

0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.010-6

10-4

10-2

100

WavefunctionWavefunction

EU0

xikxik 11 BeAex 1φ

xx DeCex αα2φ

xik1Fex 3φ

Optical AnalogOptical Analog

If second prism is brought close to the first there is a small probability for part of the incident wave to couple through the air gap and emerge in the second prism.

If reflection angle is greater than the critical angle then the light ray will be totally internally reflected

evanescent wave

Limiting CaseLimiting CaseTunnelling through wide barriers:Inside the barrier the wavefunction is proportional to exp(-x) or exp(-x/), where = 1/ is the penetration depth (see Potential step lecture). If L then very little of the wavefunction will survive to x = L. The condition for a ‘wide barrier’ is thus

1LEUm2

LL1 0

α

δ

The barrier can be considered to be wide if L is large or if E << U0.Making this approximation we see that

2e

2ee

ysinhy

1 yyy

and then 4

eysinh

y22

so for a thick barrier equation 11 reduces to

L2

00e

UE

1UE

16T α

The probability of tunnelling is then dominated by the exponentially decreasing term.

13

14

ExampleExampleAn electron (m = 9.11 10-31kg) encounters a potential barrier of height 0.100eV and width 15nm What is the transmission probability if its energy is (a) 0.040eV and (b) 0.060 eV?We first check to see if the barrier is thick (equation 13). for E

= 0.04eV

= 18.8 >> 1 thick barrier

m1015

s.J10055.1

J10602.140.010.0kg1011.92L 934

1931

δ

and for E = 0.060: L/ = 15.5 >> 1 thick barrier

we can use equation 14 for the transmission probability 168.182 108.1e

10.004.0

110.004.0

16TeV04.0E )a(

134.152 108.1e10.006.0

110.006.0

16TeV06.0E )b(

Very small in both cases!! Can we observe this in a real stuation

Field EmissionField Emission

metal

electrons bound by potential step at surface

Tunnelling through potential barrier

Cathode Anode

+V0

Field Emission Displays (FED)Field Emission Displays (FED)

Scanning Tunnelling Microscope (STM)Scanning Tunnelling Microscope (STM)

Scanning Tunnelling Microscope (STM)Scanning Tunnelling Microscope (STM)

Scanning Tunnelling Microscope (STM)Scanning Tunnelling Microscope (STM)

Pt Surface

Si (111) Surface

Pentacene molecules on Pentacene molecules on SiliconSilicon

Sample negative Sample positive

The Tunnel DiodeThe Tunnel Diodesee http://mxp.physics.umn.edu/s98/projects/menz/poster.htm

p-type

n-type

EF

EF

donors

acceptors

p-type

n-type

EF

eV0

- - + + - - + +- - + +

Conduction Band

Conduction Band

Valence Band

Valence Band

EF

eV0- eVext

p-type

n-type

- + - +- +

Vext

+ -

p-type

n-type

- - - + + +- - - + + +- - - + + +

Vext

+-

forward biased reversed biased

EF

eV0 + Vext

The Tunnel DiodeThe Tunnel Diode

Conduction Band

Valence Band

The Tunnel DiodeThe Tunnel Diode

Uranium 238 Thorium

234

Alpha particle

Strong Nuclear Force

Electrostatic repulsion

To escape the nucleus the -particle must tunnel.

Alpha Decay of NucleiAlpha Decay of Nuclei