In physics, quantum tunnelling, barrier penetration, or simply tunnelling is a quantum mechanical phenomenon in which an object such as an electron or atom passes through a potential energy barrier that, according to classical mechanics, should not be passable due to the object not having sufficient energy to pass or surmount the barrier.

Tunnelling is a consequence of the wave nature of matter and quantum indeterminacy. The quantum wave function describes the states of a particle or other physical system and wave equations such as the Schrödinger equation describe their evolution. In a system with a short, narrow potential barrier, a small part of wavefunction can appear outside of the barrier representing a probability for tunnelling through the barrier.

Since the probability of transmission of a wave packet through a barrier decreases exponentially with the barrier height, the barrier width, and the tunnelling particle's mass, tunnelling is seen most prominently in low-mass particles such as electrons tunnelling through atomically narrow barriers. However tunnelling has been observed with protons and even atoms and tunnelling has been used to explain physical effects with particles this large.

Tunnelling plays an essential role in physical phenomena such as nuclear fusion and alpha radioactive decay of atomic nuclei. Tunnelling applications include the tunnel diode, quantum computing, flash memory, and the scanning tunnelling microscope. Tunnelling limits the minimum size of devices used in microelectronics because electrons tunnel readily through insulating layers and transistors that are thinner than about 1 nm.

The effect was predicted in the early 20th century. Its acceptance as a general physical phenomenon came mid-century.

Concept

thumb|right|upright=1.5|Animation showing the tunnel effect and its application to an [[Scanning tunnelling microscope|STM]]

Quantum tunnelling falls under the domain of quantum mechanics. To understand the phenomenon, particles attempting to travel across a potential barrier 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 cannot reach the other side. Thus, a ball without sufficient energy to surmount the hill would roll back down. In quantum mechanics, a particle can, with a small probability, tunnel to the other side, thus crossing the barrier. The reason for this difference comes from treating matter as having properties of waves and particles.

Some sources describe the mere penetration of a wave function into the barrier, without transmission on the other side, as a tunnelling effect, such as in tunnelling into the walls of a finite potential well.

Tunnelling problem

thumb|A simulation of a wave packet incident on a potential barrier. In relative units, the barrier energy is 20, greater than the mean wave packet energy of 14. A portion of the wave packet passes through the barrier.|alt=

The wave function of a physical system of particles specifies everything that can be known about the system. Therefore, problems in quantum mechanics analyse the system's wave function. Using mathematical formulations, such as the Schrödinger equation, the time evolution of a known wave function can be deduced. The square of the absolute value of this wave function is directly related to the probability distribution of the particle positions, which describes the probability that the particles would be measured at those positions.

As shown in the animation, when a wave packet impinges on the barrier, most of it is reflected and some is transmitted through the barrier. The wave packet becomes more delocalised: it is now on both sides of the barrier and lower in maximum amplitude, but equal in integrated square-magnitude, meaning that the probability the particle is somewhere remains unity. The wider the barrier and the higher the barrier energy, the lower the probability of tunnelling.

Some models of a tunnelling barrier, such as the rectangular barriers shown, can be analysed and solved algebraically. Most problems do not have an algebraic solution, so numerical solutions are used. "Semiclassical methods" offer approximate solutions that are easier to compute, such as the WKB approximation.

History

The Schrödinger equation was published in 1926. The first person to apply the Schrödinger equation to a problem that involved tunnelling between two classically allowed regions through a potential barrier was Friedrich Hund in a series of articles published in 1927. He studied the solutions of a double-well potential and discussed molecular spectra. Leonid Mandelstam and Mikhail Leontovich discovered tunnelling independently and published their results in 1928.

In 1927, Lothar Nordheim, assisted by Ralph Fowler, published a paper that discussed thermionic emission and reflection of electrons from metals. He assumed a surface potential barrier that confines the electrons within the metal and showed that the electrons have a finite probability of tunnelling through or reflecting from the surface barrier when their energies are close to the barrier energy. Classically, the electron would either transmit or reflect with 100% certainty, depending on its energy. In 1928 J. Robert Oppenheimer published two papers on field emission, i.e. the emission of electrons induced by strong electric fields. Nordheim and Fowler simplified Oppenheimer's derivation and found values for the emitted currents and work functions that agreed with experiments. The latter 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. All three researchers were familiar with the works on field emission,

In the early days of quantum theory, the term tunnel effect was not used, and the effect was instead referred to as penetration of, or leaking through, a barrier. The German term wellenmechanischer Tunneleffekt was used in 1931 by Walter Schottky. In 1960, following Esaki's work, Ivar Giaever showed experimentally that tunnelling also took place in superconductors. The tunnelling spectrum gave direct evidence of the superconducting energy gap. In 1962, Brian Josephson predicted the tunnelling of superconducting Cooper pairs. Esaki, Giaever and Josephson shared the 1973 Nobel Prize in Physics for their works on quantum tunnelling in solids.

In 2025, John Clarke, John M. Martinis and Michel H. Devoret received the Nobel Prize in physics for experiments done in 1984 and 1985 that demonstrated how quantum tunnelling can be observed on a macroscopic scale, involving many particles. They built an electrical circuit with two superconductors, components that can conduct a current without any electrical resistance. They separated these with a thin layer of material that did not conduct any current at all. In this experiment, they showed that they could control and investigate a phenomenon in which all the charged particles in the superconductor behave in unison, as if they are a single particle that fills the entire circuit.

Applications

Tunnelling is used to explain some important macroscopic physical phenomena.

Solid-state physics

Electronics

Tunnelling is a source of current leakage in very-large-scale integration (VLSI) electronics and results in a substantial power drain and heating effects that plague such devices. It is considered the lower limit on how microelectronic device elements can be made. Tunnelling is a fundamental technique used to program the floating gates of flash memory.

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, and some electron microscopes.

Tunnel junction

A simple barrier can be created by separating two conductors with a very thin insulator. Tunnelling is readily detectable with potential barriers in thin-film junctions of thickness about 3 nm or smaller for electrons. Josephson junctions take advantage of quantum tunnelling and superconductivity to create the Josephson effect. This has applications in precision measurements of voltages and magnetic fields,

Because the tunnelling current drops off rapidly, tunnel diodes can be created that have a range of voltages for which current decreases as voltage increases. This peculiar property is used in some applications, such as high speed devices where the characteristic tunnelling probability changes as rapidly as the bias voltage.

Tunnel field-effect transistors

A European research project demonstrated field effect transistors in which the gate (channel) 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 would improve the performance per power of integrated circuits.

Conductivity of crystalline solids

While the Drude-Lorentz 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.

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 (an electron tunnelling into the nucleus is electron capture). This was the first application of quantum tunnelling. Radioactive decay is a relevant issue for astrobiology as this consequence of quantum tunnelling creates a constant energy source over a large time interval for environments outside the circumstellar habitable zone where insolation would not be possible (subsurface oceans) or effective.

Chemistry

Energetically forbidden reactions

Chemical reactions in the interstellar medium occur at extremely low energies. Probably the most fundamental ion-molecule reaction involves hydrogen ions with hydrogen molecules.

The quantum mechanical tunnelling rate for the same reaction using the hydrogen isotope deuterium, <chem>D- + H2 -> H- + HD</chem>, has been measured experimentally in an ion trap.

The deuterium was placed in an ion trap and cooled. The trap was then filled with hydrogen. At the temperatures used in the experiment, the energy barrier for reaction would not allow the reaction to succeed with classical dynamics alone. Quantum tunnelling allowed reactions to happen in rare collisions. It was calculated from the experimental data that collisions happened one in every hundred billion.

Kinetic isotope effect

In chemical kinetics, the substitution of a light isotope of an element with a heavier one typically results in a slower reaction rate. This is generally attributed to differences in the zero-point vibrational energies for chemical bonds containing the lighter and heavier isotopes and is generally modelled using transition state theory. However, in certain cases, large isotopic effects are observed that cannot be accounted for by a semi-classical treatment, and quantum tunnelling is required. R. P. Bell developed a modified treatment of Arrhenius kinetics that is commonly used to model this phenomenon.

Astrochemistry in interstellar clouds

By including quantum tunnelling, the astrochemical syntheses of various molecules in interstellar clouds can be explained, such as the synthesis of molecular hydrogen, water (ice) and the prebiotic important formaldehyde. Here it is important both as electron tunnelling and proton tunnelling. Electron tunnelling is a key factor in many biochemical redox reactions (photosynthesis, cellular respiration) as well as enzymatic catalysis. Proton tunnelling is a key factor in spontaneous DNA mutation. A hydrogen bond joins DNA base pairs. A double well potential along a hydrogen bond separates a potential energy barrier. It is believed that the double well potential is asymmetric, with one well deeper than the other such that the proton normally rests in the deeper well. For a mutation to occur, the proton must have tunnelled into the shallower well. The proton's movement 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.

Astrophysics

Quantum tunnelling is expected to occur at the event horizon of black holes, leading to Hawking radiation. Classically nothing can escape from a black hole, but quantum tunnelling gives a small probability for emission of radiation. More recently, experimental tunnelling time data of phonons, photons, and electrons was published by Günter Nimtz. Another experiment overseen by A. M. Steinberg, seems to indicate that particles could tunnel at apparent speeds faster than light.

Other physicists, such as Herbert Winful, disputed these claims. Winful argued that the wave packet of a tunnelling particle propagates locally, so a particle can't tunnel through the barrier non-locally. Winful also argued that the experiments that are purported to show non-local propagation have been misinterpreted. In particular, the group velocity of a wave packet does not measure its speed, but is related to the amount of time the wave packet is stored in the barrier. Moreover, if quantum tunnelling is modelled with the relativistic Dirac equation, well established mathematical theorems imply that the process is completely subluminal. It has also been proved that within a relativistic quantum field theory framework, tunnelling cannot be superluminal, even though the group velocity can be larger than the light velocity.

Dynamical tunnelling

left|thumb|upright=1|Quantum tunnelling oscillations of probability in an integrable double well of potential, seen in phase space

The concept of quantum tunnelling can be extended to situations where there exists a quantum transport between regions that are classically not connected even if there is no associated potential barrier. This phenomenon is known as dynamical tunnelling.

Tunnelling in phase space

The concept of dynamical tunnelling is particularly suited to address the problem of quantum tunnelling in high dimensions (<math>d>1</math>). In the case of an integrable system, where bounded classical trajectories are confined onto tori in phase space, tunnelling can be understood as the quantum transport between semi-classical states built on two distinct but symmetric tori.

Chaos-assisted tunnelling

left|thumb|upright=1|Chaos-assisted tunnelling oscillations between two regular tori embedded in a chaotic sea, seen in phase space

In real life, most systems are not integrable and display various degrees of chaos. Classical dynamics is then said to be mixed and the system phase space is typically composed of islands of regular orbits surrounded by a large sea of chaotic orbits. The existence of the chaotic sea, where transport is classically allowed, between the two symmetric tori then assists the quantum tunnelling between them. This phenomenon is referred as chaos-assisted tunnelling. and is characterised by sharp resonances of the tunnelling rate when varying any system parameter.

Resonance-assisted tunnelling

When <math>\hbar</math> is small in front of the size of the regular islands, the fine structure of the classical phase space plays a key role in tunnelling. In particular the two symmetric tori are coupled "via a succession of classically forbidden transitions across nonlinear resonances" surrounding the two islands.

Several phenomena have the same behaviour as quantum tunnelling. Two examples are evanescent wave coupling (the application of Maxwell's wave-equation to light) and the application of the non-dispersive wave-equation from acoustics applied to "waves on strings".

These effects are modelled similarly to the rectangular potential barrier. In these cases, 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.

A classical wave-particle association was originally analysed as analogous to quantum tunnelling, but subsequent analysis found a fluid dynamics cause related to the vertical momentum imparted to particles near the barrier.

See also

  • Proton tunnelling – Type of quantum tunnelling

References

Further reading

  • Animation, applications and research linked to tunnel effect and other quantum phenomena (Université Paris Sud)
  • Animated illustration of quantum tunnelling
  • Animated illustration of quantum tunnelling in a RTD device
  • Interactive Solution of Schrodinger Tunnel Equation