1. Quantum Tunneling & the
Scanning Tunneling Microscope
Rob Morien
Solid State Physics 551
University of Wisconsin-Milwaukee
12-16-06

2. Topics
Energy level diagrams and quantum
tunneling for ordinary metals
Energy level diagrams and quantum
tunneling for superconductors
STM

3. Energy Level Diagrams for Normal Metals
If a DV exists between both plates, electrons are
allowed to tunnel
Two requirements for tunnelling to occur:
1)Conservation of Energy
2)Electrons must tunnel to empty sates (Pauli Principal)

4. Quantum Mechanically:
Electron wave function penetrates metal
Wave function given by the Hamiltonians:
H
m
d
dx
H
m
d
dx
V
= −
⎛
⎝
⎜
⎞
⎠
⎟
= −
⎛
⎝
⎜
⎞
⎠
⎟ +
h
h
2 2
2
2 2
2
2
2
Outside metal
Inside Metal
ψ α αoutside
A x B x= +sin( ) cos( )
ψ α α
inside
i x i x
Ce De= + −

5. This implies:
The wave fxn inside the metal has an
exponential that goes to infinity and can be
neglected.
The real part decays exponentially.
Since there is a non-zero wave fxn, there
remains a probability that an electron can
tunnel into the metal.

6. Common Superconductor Arrangements
and Materials
Superconductor &
Normal Metal
Superconductor &
Superconductor (both
same materials)
Superconductor &
Superconductor (both
different materials)
Indium
Tin
Mercury
Vanadium
Lead

7. Energy Level Diagram for a Superconductor
E required
To split
Electrons:
( ){ }
E E E
E
P
m
P
m
P
m
P
m
i f
i f
i f j f
= +
= − +
= −
⎛
⎝
⎜
⎞
⎠
⎟ +
⎧
⎨
⎩
⎫
⎬
⎭
+ −
⎛
⎝
⎜
⎞
⎠
⎟ +
⎧
⎨
⎩
⎫
⎬
⎭
ε ε
2
2
1
2
2
2
1
2 2
2
1
2
2 2 2 2
Δ
Δ Δ
Δ =
=
− =
= − −
−
2
1
h e
AveragePhononFrequency
V MatrixElementoftheScatteringInteraction
N Density of States
L
N V
L
F
F
ν
ν
ε
ε( )
( )

8. Tunneling between normal metals and
Superconductors
Positive bias placed on S.C.
Tunneling can occur once has been reached
means bottom of E-level for quasi-particles
has passed fermi-level of normal metal
Δ / e
Δ / e

9. Graph of Current Flow for both bias’s
Negative bias placed on S.C.
Tunneling can occur once - has been reachedΔ / e

10. Tunneling between two superconductors
of like materials
With no bias, tunneling is allowed, but current is
equal for both s.c.’s, therefore no net e flow
Positive bias on left s.c. causes e’s to flow from
right to left, but not left to right.

11. Increasing to voltage
Tunneling continues until V reaches
where the electron pair is allowed to split.
One particle tunnels to left hand s.c. and one
particle tunnels to right hand s.c.

12. I-V Curves for both types of s.c.’s