Quantum tunnelling or tunneling refers to
the quantum mechanical phenomenon where a
particle tunnels through a barrier that it
classically could not surmount. This plays
an essential role in several physical phenomena,
such as the nuclear fusion that occurs in
main sequence stars like the Sun. It has important
applications to modern devices such as the
tunnel diode, quantum computing, and the scanning
tunnelling microscope. The effect was predicted
in the early 20th century and its acceptance,
as a general physical phenomenon, came mid-century.
Tunnelling is often explained using the Heisenberg
uncertainty principle and the wave–particle
duality of matter. Pure quantum mechanical
concepts are central to the phenomenon, so
quantum tunnelling is one of the novel implications
of quantum mechanics.
History
Quantum tunnelling was developed from the
study of radioactivity, which was discovered
in 1896 by Henri Becquerel. Radioactivity
was examined further by Marie and Pierre Curie,
for which they earned the Nobel Prize in Physics
in 1903. Ernest Rutherford and Egon Schweidler
studied its nature, which was later verified
empirically by Friedrich Kohlrausch. The idea
of the half-life and the impossibility of
predicting decay was created from their work.
Friedrich Hund was the first to take notice
of tunnelling in 1927 when he was calculating
the ground state of the double-well potential.
Its first application was a mathematical explanation
for alpha decay, which was done in 1928 by
George Gamow and independently by Ronald Gurney
and Edward Condon. The two researchers simultaneously
solved the Schrödinger equation for a model
nuclear potential and derived a relationship
between the half-life of the particle and
the energy of emission that depended directly
on the mathematical probability of tunnelling.
After attending a seminar by Gamow, Max Born
recognised the generality of tunnelling. He
realised that it was not restricted to nuclear
physics, but was a general result of quantum
mechanics that applies to many different systems.
Shortly thereafter, both groups considered
the case of particles tunnelling into the
nucleus. The study of semiconductors and the
development of transistors and diodes led
to the acceptance of electron tunnelling in
solids by 1957. The work of Leo Esaki, Ivar
Giaever and Brian Josephson predicted the
tunnelling of superconducting Cooper pairs,
for which they received the Nobel Prize in
Physics in 1973.
Introduction to the concept
Quantum tunnelling falls under the domain
of quantum mechanics: the study of what happens
at the quantum scale. This process cannot
be directly perceived, but much of its understanding
is shaped by the microscopic world, which
classical mechanics cannot adequately explain.
To understand the phenomenon, particles attempting
to travel between potential barriers can be
compared to a ball trying to roll over a hill;
quantum mechanics and classical mechanics
differ in their treatment of this scenario.
Classical mechanics predicts that particles
that do not have enough energy to classically
surmount a barrier will not be able to reach
the other side. Thus, a ball without sufficient
energy to surmount the hill would roll back
down. Or, lacking the energy to penetrate
a wall, it would bounce back or in the extreme
case, bury itself inside the wall. In quantum
mechanics, these particles can, with a very
small probability, tunnel to the other side,
thus crossing the barrier. Here, the "ball"
could, in a sense, borrow energy from its
surroundings to tunnel through the wall or
"roll over the hill", paying it back by making
the reflected electrons more energetic than
they otherwise would have been.
The reason for this difference comes from
the treatment of matter in quantum mechanics
as having properties of waves and particles.
One interpretation of this duality involves
the Heisenberg uncertainty principle, which
defines a limit on how precisely the position
and the momentum of a particle can be known
at the same time. This implies that there
are no solutions with a probability of exactly
zero, though a solution may approach infinity
if, for example, the calculation for its position
was taken as a probability of 1, the other,
i.e. its speed, would have to be infinity.
Hence, the probability of a given particle's
existence on the opposite side of an intervening
barrier is non-zero, and such particles will
appear on the 'other' side with a relative
frequency proportional to this probability.
The tunnelling problem
The wave function of a particle summarises
everything that can be known about a physical
system. Therefore, problems in quantum mechanics
center around the analysis of the wave function
for a system. Using mathematical formulations
of quantum mechanics, such as the Schrödinger
equation, the wave function can be solved.
This is directly related to the probability
density of the particle's position, which
describes the probability that the particle
is at any given place. In the limit of large
barriers, the probability of tunnelling decreases
for taller and wider barriers.
For simple tunnelling-barrier models, such
as the rectangular barrier, an analytic solution
exists. Problems in real life often do not
have one, so "semiclassical" or "quasiclassical"
methods have been developed to give approximate
solutions to these problems, like the WKB
approximation. Probabilities may be derived
with arbitrary precision, constrained by computational
resources, via Feynman's path integral method;
such precision is seldom required in engineering
practice.
Related phenomena
There are several phenomena that have the
same behaviour as quantum tunnelling, and
thus can be accurately described by tunnelling.
Examples include the tunnelling of a classical
wave-particle association, evanescent wave
coupling and the application of the non-dispersive
wave-equation from acoustics applied to "waves
on strings". Evanescent wave coupling, until
recently, was only called "tunnelling" in
quantum mechanics; now it is used in other
contexts.
These effects are modelled similarly to the
rectangular potential barrier. In these cases,
there is one transmission medium through which
the wave propagates that is the same or nearly
the same throughout, and a second medium through
which the wave travels differently. This can
be described as a thin region of medium B
between two regions of medium A. The analysis
of a rectangular barrier by means of the Schrödinger
equation can be adapted to these other effects
provided that the wave equation has travelling
wave solutions in medium A but real exponential
solutions in medium B.
In optics, medium A is a vacuum while medium
B is glass. In acoustics, medium A may be
a liquid or gas and medium B a solid. For
both cases, medium A is a region of space
where the particle's total energy is greater
than its potential energy and medium B is
the potential barrier. These have an incoming
wave and resultant waves in both directions.
There can be more mediums and barriers, and
the barriers need not be discrete; approximations
are useful in this case.
Applications
Tunnelling occurs with barriers of thickness
around 1-3 nm and smaller, but is the cause
of some important macroscopic physical phenomena.
For instance, tunnelling is a source of current
leakage in very-large-scale integration electronics
and results in the substantial power drain
and heating effects that plague high-speed
and mobile technology; it is considered the
lower limit on how small computer chips can
be made.
Radioactive decay
Radioactive decay is the process of emission
of particles and energy from the unstable
nucleus of an atom to form a stable product.
This is done via the tunnelling of a particle
out of the nucleus. This was the first application
of quantum tunnelling and led to the first
approximations.
Spontaneous DNA mutation
Spontaneous mutation of DNA occurs when normal
DNA replication takes place after a particularly
significant proton has defied the odds in
quantum tunnelling in what is called "proton
tunnelling". A hydrogen bond joins normal
base pairs of DNA. There exists a double well
potential along a hydrogen bond separated
by a potential energy barrier. It is believed
that the double well potential is asymmetric
with one well deeper than the other so the
proton normally rests in the deeper well.
For a mutation to occur, the proton must have
tunnelled into the shallower of the two potential
wells. The movement of the proton from its
regular position is called a tautomeric transition.
If DNA replication takes place in this state,
the base pairing rule for DNA may be jeopardised
causing a mutation. Per-Olov Lowdin was the
first to develop this theory of spontaneous
mutation within the double helix. Other instances
of quantum tunnelling-induced mutations in
biology are believed to be a cause of ageing
and cancer.
Cold emission
Cold emission of electrons is relevant to
semiconductors and superconductor physics.
It is similar to thermionic emission, where
electrons randomly jump from the surface of
a metal to follow a voltage bias because they
statistically end up with more energy than
the barrier, through random collisions with
other particles. When the electric field is
very large, the barrier becomes thin enough
for electrons to tunnel out of the atomic
state, leading to a current that varies approximately
exponentially with the electric field. These
materials are important for flash memory,
vacuum tubes, as well as some electron microscopes.
Tunnel junction
A simple barrier can be created by separating
two conductors with a very thin insulator.
These are tunnel junctions, the study of which
requires quantum tunnelling. Josephson junctions
take advantage of quantum tunnelling and the
superconductivity of some semiconductors to
create the Josephson effect. This has applications
in precision measurements of voltages and
magnetic fields, as well as the multijunction
solar cell.
Tunnel diode
Diodes are electrical semiconductor devices
that allow electric current flow in one direction
more than the other. The device depends on
a depletion layer between N-type and P-type
semiconductors to serve its purpose; when
these are very heavily doped the depletion
layer can be thin enough for tunnelling. Then,
when a small forward bias is applied the current
due to tunnelling is significant. This has
a maximum at the point where the voltage bias
is such that the energy level of the p and
n conduction bands are the same. As the voltage
bias is increased, the two conduction bands
no longer line up and the diode acts typically.
Because the tunnelling current drops off rapidly,
tunnel diodes can be created that have a range
of voltages for which current decreases as
voltage is increased. This peculiar property
is used in some applications, like high speed
devices where the characteristic tunnelling
probability changes as rapidly as the bias
voltage.
The resonant tunnelling diode makes use of
quantum tunnelling in a very different manner
to achieve a similar result. This diode has
a resonant voltage for which there is a lot
of current that favors a particular voltage,
achieved by placing two very thin layers with
a high energy conductance band very near each
other. This creates a quantum potential well
that have a discrete lowest energy level.
When this energy level is higher than that
of the electrons, no tunnelling will occur,
and the diode is in reverse bias. Once the
two voltage energies align, the electrons
flow like an open wire. As the voltage is
increased further tunnelling becomes improbable
and the diode acts like a normal diode again
before a second energy level becomes noticeable.
Tunnel field-effect transistors
A European research project has demonstrated
field effect transistors in which the gate
is controlled via quantum tunnelling rather
than by thermal injection, reducing gate voltage
from ~1 volt to 0.2 volts and reducing power
consumption by up to 100×. If these transistors
can be scaled up into VLSI chips, they will
significantly improve the performance per
power of integrated circuits.
Quantum conductivity
While the Drude model of electrical conductivity
makes excellent predictions about the nature
of electrons conducting in metals, it can
be furthered by using quantum tunnelling to
explain the nature of the electron's collisions.
When a free electron wave packet encounters
a long array of uniformly spaced barriers
the reflected part of the wave packet interferes
uniformly with the transmitted one between
all barriers so that there are cases of 100%
transmission. The theory predicts that if
positively charged nuclei form a perfectly
rectangular array, electrons will tunnel through
the metal as free electrons, leading to an
extremely high conductance, and that impurities
in the metal will disrupt it significantly.
Scanning tunnelling microscope
The scanning tunnelling microscope, invented
by Gerd Binnig and Heinrich Rohrer, allows
imaging of individual atoms on the surface
of a metal. It operates by taking advantage
of the relationship between quantum tunnelling
with distance. When the tip of the STM's needle
is brought very close to a conduction surface
that has a voltage bias, by measuring the
current of electrons that are tunnelling between
the needle and the surface, the distance between
the needle and the surface can be measured.
By using piezoelectric rods that change in
size when voltage is applied over them the
height of the tip can be adjusted to keep
the tunnelling current constant. The time-varying
voltages that are applied to these rods can
be recorded and used to image the surface
of the conductor. STMs are accurate to 0.001 nm,
or about 1% of atomic diameter.
Faster than light
It is possible for spin zero particles to
travel faster than the speed of light when
tunnelling. This apparently violates the principle
of causality, since there will be a frame
of reference in which it arrives before it
has left. However, careful analysis of the
transmission of the wave packet shows that
there is actually no violation of relativity
theory. In 1998, Francis E. Low reviewed briefly
the phenomenon of zero time tunnelling. More
recently experimental tunnelling time data
of phonons, photons, and electrons have been
published by Günter Nimtz.
Mathematical discussions of quantum tunnelling
The following subsections discuss the mathematical
formulations of quantum tunnelling.
The Schrödinger equation
The time-independent Schrödinger equation
for one particle in one dimension can be written
as
or
where is the reduced Planck's constant, m
is the particle mass, x represents distance
measured in the direction of motion of the
particle, Ψ is the Schrödinger wave function,
V is the potential energy of the particle,
E is the energy of the particle that is associated
with motion in the x-axis, and M(x) is a quantity
defined by V(x) - E which has no accepted
name in physics.
The solutions of the Schrödinger equation
take different forms for different values
of x, depending on whether M(x) is positive
or negative. When M(x) is constant and negative,
then the Schrödinger equation can be written
in the form
The solutions of this equation represent traveling
waves, with phase-constant +k or -k. Alternatively,
if M(x) is constant and positive, then the
Schrödinger equation can be written in the
form
The solutions of this equation are rising
and falling exponentials in the form of evanescent
waves. When M(x) varies with position, the
same difference in behaviour occurs, depending
on whether M(x) is negative or positive. It
follows that the sign of M(x) determines the
nature of the medium, with positive M(x) corresponding
to medium A as described above and negative
M(x) corresponding to medium B. It thus follows
that evanescent wave coupling can occur if
a region of positive M(x) is sandwiched between
two regions of negative M(x), hence creating
a potential barrier.
The mathematics of dealing with the situation
where M(x) varies with x is difficult, except
in special cases that usually do not correspond
to physical reality. A discussion of the semi-classical
approximate method, as found in physics textbooks,
is given in the next section. A full and complicated
mathematical treatment appears in the 1965
monograph by Fröman and Fröman noted below.
Their ideas have not been incorporated into
physics textbooks, but their corrections have
little quantitative effect.
The WKB approximation
The wave function is expressed as the exponential
of a function:
, where
is then separated into real and imaginary
parts:
, where A(x) and B(x) are real-valued functions.
Substituting the second equation into the
first and using the fact that the imaginary
part needs to be 0 results in:
.
To solve this equation using the semiclassical
approximation, each function must be expanded
as a power series in . From the equations,
the power series must start with at least
an order of to satisfy the real part of the
equation; for a good classical limit starting
with the highest power of Planck's constant
possible is preferable, which leads to
and
,
with the following constraints on the lowest
order terms,
and
.
At this point two extreme cases can be considered.
Case 1 If the amplitude varies slowly as compared
to the phase and
which corresponds to classical motion. Resolving
the next order of expansion yields
Case 2
If the phase varies slowly as compared to
the amplitude, and
which corresponds to tunnelling. Resolving
the next order of the expansion yields
In both cases it is apparent from the denominator
that both these approximate solutions are
bad near the classical turning points . Away
from the potential hill, the particle acts
similar to a free and oscillating wave; beneath
the potential hill, the particle undergoes
exponential changes in amplitude. By considering
the behaviour at these limits and classical
turning points a global solution can be made.
To start, choose a classical turning point,
and expand in a power series about :
Keeping only the first order term ensures
linearity:
.
Using this approximation, the equation near
becomes a differential equation:
.
This can be solved using Airy functions as
solutions.
Taking these solutions for all classical turning
points, a global solution can be formed that
links the limiting solutions. Given the 2
coefficients on one side of a classical turning
point, the 2 coefficients on the other side
of a classical turning point can be determined
by using this local solution to connect them.
Hence, the Airy function solutions will asymptote
into sine, cosine and exponential functions
in the proper limits. The relationships between
and are
and
With the coefficients found, the global solution
can be found. Therefore, the transmission
coefficient for a particle tunnelling through
a single potential barrier is
,
where are the 2 classical turning points for
the potential barrier.
See also
Dielectric barrier discharge
Field electron emission
Holstein–Herring method
Superconducting tunnel junction
Tunnel diode
Tunnel junction
References
Further reading
N. Fröman and P.-O. Fröman. JWKB Approximation:
Contributions to the Theory. Amsterdam: North-Holland. 
Razavy, Mohsen. Quantum Theory of Tunneling.
World Scientific. ISBN 981-238-019-1. 
Griffiths, David J.. Introduction to Quantum
Mechanics. Prentice Hall. ISBN 0-13-805326-X. 
James Binney and Skinner, D.. The Physics
of Quantum Mechanics: An Introduction. Cappella
Archive. ISBN 1-902918-51-7. 
Liboff, Richard L.. Introductory Quantum Mechanics.
Addison-Wesley. ISBN 0-8053-8714-5. 
Vilenkin, Alexander; Vilenkin, Alexander;
Winitzki, Serge. "Particle creation in a tunneling
universe". Physical Review D 68: 023520. arXiv:gr-qc/0210034.
Bibcode:2003PhRvD..68b3520H. doi:10.1103/PhysRevD.68.023520. 
External links
Animation, applications and research linked
to tunnel effect and other quantum phenomena
Animated illustration of quantum tunnelling
Animated illustration of quantum tunnelling
in a RTD device
