Some applications of Quantum Tunneling

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Some applications of Quantum Tunneling
Scanning tunneling microscope (STM)
Field emission
? Emission in Nuclear Decay
Ammonia inversion
Decay of Black holes
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Binnig
Rohrer
Scanning Tunneling Microscope
Due to the quantum effect of barrier penetration,
the electron density of a material extends
beyond its surface
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Needle tip of couple atoms Samples a region of
0.2 nm in size
Surface electrons in a metal tunnel to needle tip
One can exploit this to measure the electron
density on a materials surface
Circular ring of radius 7.13 nm built from 48 Fe
atoms deposited by STM on a Cu crystal surface
Can see interference pattern created by surface
electrons scattering off Fe cage
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With no probe, wave function dies off
exponentially from surface of metal
With a probe, wave function falls off
exponentially from surface of metal, but has
finite probability to penetrate barrier and
produce current in the probe
extremely sensitive to L
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Tunneling current is extremely sensitive to L
(L gt 0)
Assuming that L is changed by 0.01 nm
0.90
Assuming that L is changed by 0.1 nm
0.37
STM resolution better than 0.01 nm in height !
Cf, SEM resolution 2 nm, Optical resolution
200 nm
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A barrier of arbitrary shape can be approximated
as a succession of square barriers
E lt U(x)
classically forbidden region
E
Wave function in a square barrier (for energy E lt
U0)
x1
x2
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Electron emission
How to let electrons escape from a metal ?
thermionic emission kBT
metal
work function W
e-
Electrons bound to a metal are literally torn
from the surface by the application of a strong
electric field
potential barrier
Anode
Cathode
FED (Field emission display), monitor
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Electric field ?
Suppose we create an electric field that attracts
free electrons away from the metal to a nearby
anode Force Fe attracts free electrons
metal
W
U(x) W ? e ? x (U0W )
(e electronic charge)
e-
E 0 (EF)
x2
U(x) electric potential outside the metal
(inside the metal, U 0)
x0
Triangular shape potential barrier in regime
between x 0 and x2 W / e ?
The electron can tunnel through the triangular
potential barrier !
(12/9/2009)
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Tunneling probability
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Tunneling probability
Define a critical field
5?1010 V/m, for W 4 eV
Many electrons in the metal lead to many
collisions with the wall (per area per second)
1030 /s
Emission rate f ?T
As a result, detectable currents pA are
possible even for ? ?c/50
Field emission microscope can have a
magnification of 3,000,000?. Individual atoms can
be seen using this technique
3 ? 10-11 Amp
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? Emission in Nuclear Decay
1910, Rutherford found ? particles emitted from
212Po
E? 8.95 MeV, ? 3.0?10-7 sec
E? 4.05 MeV, ? 1.4?1010 year
half-life of the emitter
Two puzzling aspects of ? particle decay All ?
particles emitted from any one source have nearly
the same KE a few MeV for all known
emitters The half-life of the emitter ranges
over 24 orders of magnitude
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• Inside the nuclear (Ro 10 fm), ? particles are
  bound by the strong, short-range nuclear force
• Outside the nuclear, the potential is due to
  Coulomb repulsion between the (Z-2) protons
  remaining in nucleus and the 2 protons of ?
  particle

Potential seen by ? particle
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Ucoulomb(R0 10 fm) 30 MeV
Let
E?
In the region Ro lt r lt R1, ? particle does not
have enough energy to escape classically, but it
can tunnel out quantum-mechanically
R1
Ro
1928, Gamow, Gurney, Gurney
Tunneling probability
Solving integral approximately R1 gtgt Ro
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fine structure constant
Escape probability in one encounter with barrier
Decay rate Each time it reaches the barrier,
there is a probability T of escaping, so the
decay rate ( of decays / time)
( of times that ? strikes the barrier per
second) ? T
Half-life time
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Therefore,

• Nuclear decay to ? particle
• can be well described as
• quantum tunneling through
• a Coulomb barrier
• Wide range in lifetime
• can be explained by the
• exponential transmission
• probability

Log10(? )
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Ammonia molecule (NH3) Ammonia maser
tunneling of an entire atom, but not just of a
single electron !!
N
The N atom in the ammonia molecule can have two
equal configurations on both sides of the H-plane
(Looking along the H-plane)
The N atom in the ammonia molecule (NH3)
constitutes a double oscillator
Due to the quantum tunneling effect, the molecule
flip-flops repeatedly at a rate of 2.4?1010 Hz
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Spatially symmetric potential
a
-a
Overlap reduces potential at x 0 and increases
chance of tunneling
The two lowest eigenstates
yodd
antisymmetric
yeven
symmetric
The states are superpositions of these two
single-well wavefunctions
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Townes and Gordon NH3 maser (1953)
Stimulated Emission of radiation between these
two lowest energy configurations of ammonia
creates the Ammonia Maser E1 46.37 meV,
E2 46.55 meV DE E2 E1 0.18 meV
?E is small and population inversion can be
readily produced
What wavelength of radiation does the maser emit?
where Ephoton hc/l
By energy conservation, E2 - E1 Ephoton,
l hc/(E2 - E1) 1240 eVnm/1.8x10-4 eV 6888
mm 6.888 mm
Microwave Laser! (Maser)
MASER Microwave Amplification by Stimulated
Emission of Radiation LASER Light Amplification
by Stimulated Emission of Radiation
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Interaction between radiation and atom
The process of (a) absorption, and (b)
spontaneous emission. The lifetime of the upper
state is ts, and the photon is emitted in a
random direction. (c) Stimulated emission. In
this process, the emitted photons vibrate in
phase with the stimulating photon, and all have
the same direction of travel. The emitted and the
stimulating photons are coherent
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The periodic inversion of the N atom in NH3 the
oscillations of the N atom between the two minima
occur at f 2.3786?1010 Hz when the molecule
is in the ground state, which is much lower than
the vibrations without barrier penetration. Due
to its relatively low but fundamental value, this
vibration was used for atomic clocks (a very
stable oscillator)
Ammonia molecules are used as an ultrasensitive
regulator of the frequency of the microwave
frequency transmitter
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Hawking
Decay of Black holes
Most of what we believe about black holes has
come from theoretical models in general
relativity. Once inside the event horizon,
nothing even light, can escape from gravitational
pull of a black hole
If a black hole traps all the light that crosses
the event horizon, then how can we ever hope to
observe one?
Quantum mechanical tunneling
Hawking (1974) proposed that black holes are
radiant objects, emitting a variety of particles
by a mechanism involving tunneling through the
gravitational potential barrier surrounding the
black hole ? Black hole evaporation (the death
of a black hole)
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Applications of Tunneling Resonant Tunneling
Transistor
For very small changes in the gate voltage, the
quantized level goes in and out of resonance with
the electron energy, causing a large change in
the current in the device (or the voltage across
an external resistor)
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Summary The step potential (Reflection,
Transmission, penetration depth) The square
barrier potential (resonant tunneling) Applicatio
ns of tunneling effect
(12/14/2009)