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Tunneling Applications

Outline

• Barrier Reflection and Penetration
• Electron Conduction
• Scanning Tunneling Microscopy
• Flash Memory

A Simple

Potential Step

Region 1 Region 2

CASE I : Eo > V

In Region 1:

In Region 2:

A =^ Ae

jk 1 x

E = Eo

E = 0

x = 0

x

Eo =

~^2

2 m

@^2

@x^2

(Eo V ) =

~^2

2 m

@^2

@x^2

k

2 1 =^

2 mEo

~^2

k

2 2 =^

2 m (Eo V )

~^2

V

jk x B Be^ y =^1

jk x C Ce^ y = -^2

A Simple

Potential Step

CASE I : Eo > V

B

A

1 k 2 /k 1

1 + k 2 /k 1

k 1 k 2

k 1 + k 2

C

A

1 + k 2 /k 1

2 k 1

k 1 + k 2

A + B = C

A B =

k 2

k 1

C

Region 1 Region 2

A =^ Ae

jk 1 x

E = Eo

E = 0

x = 0

x

V

jk x B Be^ y =^1

jk x C Ce^ y = -^2

Quantum Electron Currents

Given an electron of mass

that is located in space with charge density

and moving with momentum corresponding to

… then the current density for a single electron is given by

m

⇢ = q | (x)|

2

< p > < v >=^ ~k/m

J = ⇢v = q | |

2 (~k/m)

A Simple

Potential Step

CASE I : Eo > V

1

1

Reflection = R =

B

A

2

k 1 k 2

k 1 + k 2

2

Transmission = T = 1 R

4 k 1 k 2

|k 1 + k 2 |

2

T

R

T + R = 1

k 2 k 1

=

r 1

V Eo = V Eo = 1 Eo

Region 1 Region 2

A =^ Ae

jk 1 x

E = 0

x = 0

x

V

jk x B Be^ y =^1

jk x C Ce^ y = -^2

A Simple

Potential Step

CASE II : Eo < V

In Region 1:

In Region 2:

Eo =

~^2

2 m

@^2

@x^2

(Eo V ) =

~^2

2 m

@^2

@x^2

k

2 1 =^

2 mEo

~^2

2

2 m (Eo V )

~^2

C =^ Ce

x

Region 1 Region 2

A =^ Ae

jk 1 x

E = Eo E = 0

x = 0

x

V

jk x B Be^ y =^1

A Simple

Potential Step

CASE II : Eo < V

Total reflection à Transmission must be zero

B

A

1 + j/k 1

1 j/k 1

C

A

1 j/k 1

R =

B

A

2 = 1 T^ = 0

A + B = C

A B = j

k 1

C

C =^ Ce

x

Region 1 Region 2

A =^ Ae

jk 1 x

E = Eo E = 0

x = 0

x

V

jk x B Be^ y =^1

Quantum Tunneling Through a Thin Potential Barrier

Total Reflection at Boundary

Frustrated Total Reflection (Tunneling)

2 a = L

R = 1 T^ = 0

T 6 = 0

A Rectangular

Potential Step

x=0 x=L

2 a = L

Tunneling Applet: http://www.colorado.edu/physics/phet/dev/quantum-tunneling/1.07.00/

Real part of Ψ for Eo < V , shows hyperbolic (exponential) decay in the barrier domain and decrease in amplitude of the transmitted wave.

Transmission Coefficient versus Eo/V for barrier with

for Eo < V : T =

F

A

2

1 + 14 V^

2 Eo(V Eo) sinh

2 (2a)

sinh

2 (2a) =

e

2 a e

2 a⇤^2 ⇡ e

4 a

• Normally, the car can only get as far as B,

before it falls back again

• But a fluctuation in energy could get it over the barrier to E!
• A particle
  borrows an energy Δ E to get over a barrier
• Does not violate the uncertainty principle,

provided this energy is repaid within a certain time Δ t

Imagine the Roller Coaster ...

Et

h 4 ⇡

Start position with zero speed

A

B

Multiple Choice Questions

Consider a particle tunneling through a barrier:

1. Which of the following will increase the

likelihood of tunneling?

a. decrease the height of the barrier b. decrease the width of the barrier c. decrease the mass of the particle

2. What is the energy of the particles that have successfully escaped?

a. < initial energy b. = initial energy c. > initial energy

0 L

V

x

Eo

Although the amplitude of the wave is smaller after the barrier, no energy is lost in the tunneling process

Flash Memory

Erased  1 

Stored Electrons

Programmed  0 

Insulating Dielectric

SOURCE

FLOATING GATE

CONTROL GATE

CHANNEL

Tunnel Oxide

Substrate

Channel

Floating Gate

Electrons tunnel preferentially when a voltage is applied

DRAIN

x

V (x)

Image is in the public domain

UNPROGRAMMED PROGRAMMED

To obtain the same channel charge, the programmed gate needs a higher control-gate voltage than the unprogrammed gate

Reading Flash Memory

How do we WRITE Flash Memory?

CONTROL GATE

FLOATING GATE

SILICON

CONTROL GATE

FLOATING GATE

Reading a bit means:

1. Apply Vread on the control gate
2. Measure drain current Id of the floating-gate transistors

Reading Flash Memory

VT = Q/Cpp

Vread Vgs

Id

(^00100)! Iread 0 (^00000)! I read = 0