Quantum tunnelling
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
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
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.
A great success of the tunnelling theory was the mathematical explanation for alpha decay, which was developed in 1928 by George Gamow and independently by Ronald Gurney and Edward Condon. 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, and Gamow was aware of Mandelstam and Leontovich's findings.
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. The English term tunnel effect entered the language in 1932 when it was used by Yakov Frenkel in his textbook.
In 1957 Leo Esaki demonstrated tunnelling of electrons over a few nanometre wide barrier in a semiconductor structure and developed a diode based on the tunnel effect. 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 1981, Gerd Binnig and Heinrich Rohrer developed a new type of microscope, called scanning tunnelling microscope, which is based on tunnelling and is used for imaging surfaces at the atomic level. Binnig and Rohrer were awarded the Nobel Prize in Physics in 1986 for their discovery.
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 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, as well as the multijunction solar cell.Tunnel diode
s 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 heavily doped the depletion layer can be thin enough for tunnelling. 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 valence electrons in the P-side and conduction-band electrons of the N-side are the same. As the voltage bias is increased, the two energy 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 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.
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 a current favours a particular voltage, achieved by placing two thin layers with a high energy conductance band near each other. This creates a quantum potential well that has a discrete lowest energy level. When this energy level is higher than that of the electrons, no tunnelling occurs and the diode is in reverse bias. Once the two voltage energies align, the electrons flow like an open wire. As the voltage further increases, tunnelling becomes improbable and the diode acts like a normal diode again before a second energy level becomes noticeable.