# Tunneling (PPT - 6.4MB) - MIT OpenCourseWare

Software and s/w Development

Oct 30, 2013 (4 years and 6 months ago)

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Tunneling

Outline

-

Review: Barrier Reflection

-

Barrier Penetration (Tunneling)

-

Flash Memory

A Simple

Potential Step

Region 1

Region 2

CASE I :
E
o

> V

In Region 1:

In Region 2:

A Simple

Potential Step

CASE I :
E
o

> V

is continuous:

is continuous:

Region 1

Region 2

A Simple

Potential Step

CASE I :
E
o

> V

Region 1

Region 2

Example from:
-
phet/one
-
at
-
a
-
time

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

A Simple

Potential Step

CASE I :
E
o

> V

Region 1

Region 2

A Simple

Potential Step

CASE I :
E
o

> V

1

1

Region 1

Region 2

Image originally created by the IBM Corporation.

http://ocw.mit.edu/fairuse
.

Image originally created by the IBM Corporation.

http://ocw.mit.edu/fairuse
.

Image originally created by the IBM Corporation.

http://ocw.mit.edu/fairuse
.

A Simple

Potential Step

CASE II :
E
o

< V

In Region 1:

In Region 2:

Region 1

Region 2

A Simple

Potential Step

is continuous:

is continuous:

CASE II :
E
o

< V

Region 1

Region 2

A Simple

Potential Step

CASE II :
E
o

< V

Total reflection

Transmission must be zero

Region 1

Region 2

2
a = L

Quantum Tunneling
Through a Thin Potential Barrier

Total Reflection at Boundary

Frustrated Total Reflection (Tunneling)

CASE II :
E
o

< V

Region 1

Region 2

Region 3

In Regions 1 and 3:

In Region 2:

A Rectangular

Potential Step

for E
o

< V :

A Rectangular

Potential Step

x=0

x=L

2
a = L

Tunneling Applet
:
-
tunneling/1.07.00/

Real part of Ψ for
E
o

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

Transmission Coefficient versus E
o
/V

for barrier with

for E
o

< V :

Electrons tunnel preferentially when a voltage is applied

Flash Memory

Erased

1

Stored
Electrons

Programmed

0

Insulating
Dielectric

SOURCE

FLOATING GATE

CONTROL GATE

CHANNEL

Tunnel Oxide

Substrate

Channel

Floating
Gate

DRAIN

Image is in the public domain

MOSFET: Transistor in a Nutshell

Tunneling causes thin insulating layers
to become leaky !

Conducting Channel

Image is in the public domain

Conduction electron flow

Control Gate

Semiconductor

Image courtesy of J. Hoyt Group, EECS, MIT.

Photo by L. Gomez

Image courtesy of J. Hoyt Group, EECS, MIT.

Photo by L. Gomez

UNPROGRAMMED

PROGRAMMED

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

How do we WRITE Flash Memory ?

CONTROL GATE

FLOATING GATE

SILICON

CONTROL GATE

FLOATING GATE

0

L

V
0

x

E
o

metal

metal

air

gap

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

Example: Barrier Tunneling

Let

s consider a tunneling problem:

An electron with a total energy of
E
o
= 6 eV
approaches a potential barrier with a height of

V
0

= 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?

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

Multiple Choice Questions

0

L

V

x

E
o

Although the
amplitude

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

Application of Tunneling:

Scanning
Tunneling Microscopy (STM)

Due to the quantum effect of

barrier penetration,

the
electron density of a material extends beyond its surface:

material

STM tip

~ 1 nm

One can exploit this
to measure the
electron density on a
material

s s畲晡ce:

Sodium atoms

on metal:

STM images

Single walled

carbon nanotube:

V

E
0

STM tip

material

Image originally created

by IBM Corporation

Image is in the public domain

http://ocw.mit.edu/fairuse
.

Reflection of EM Waves and QM Waves

= probability of a particular

photon being reflected

= probability of a particular

electron being reflected

Then for optical material when μ=μ
0
:

MIT OpenCourseWare

http://ocw.mit.edu

6.007 Electromagnetic Energy: From Motors to Lasers

Spring 2011