# Tunnel Effect

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## tunnel effect

[′tən·əl i‚fekt]## Tunnel Effect

the passage of a particle through a potential barrier when the particle’s total energy (which remains unchanged in the tunnel effect) is less than the height of the barrier. The tunnel effect is a quantum-mechanical phenomenon and is impossible according to classical mechanics. It is analogous to the penetration, in wave optics, of a light wave into a reflecting medium (to distances of the order of the wavelength of the light) when total internal reflection occurs from the standpoint of geometrical optics. The tunnel effect underlies many important processes in such fields as atomic, molecular, nuclear, and solid-state physics.

The tunnel effect can be explained by reference to the uncertainty relation (*see*QUANTUM MECHANICS, UNCERTAINTY PRINCIPLE, and WAVE-PARTICLE DUALITY). A classical particle cannot be located inside a potential barrier of height *V* if the particle’s energy *E* is less than *V*, since the kinetic energy of the particle *p ^{2}/2m = E - V* (where

*m*is the mass of the particle) would be negative and the particle’s momentum

*p*would be an imaginary quantity. For a particle on the microscopic scale, however, this conclusion is invalid: because of the uncertainty relation, fixing the particle in a region of space within the barrier makes the particle’s momentum indeterminate. Consequently, there is a nonzero probability of finding the particle in a region that is forbidden from the standpoint of classical mechanics. Accordingly, there is a definite probability of the particle’s passage through the potential barrier. This circumstance corresponds to the tunnel effect. The probability increases as the mass of the particle decreases, the thickness of the potential barrier decreases, and the difference

*V - E*between the particle’s energy and the height of the barrier decreases. The probability of passage through the barrier is called the penetration probability and is the principal factor determining the physical characteristics of the tunnel effect.

In the case of a one-dimensional potential barrier, the tunnel effect is characterized by the barrier’s transmission coefficient, which is equal to the ratio of the flux of particles passing through the barrier and the flux incident on the barrier. In the case of a three-dimensional potential barrier bounding a closed region of space with reduced potential energy (a potential well), the tunnel effect is characterized by the probability *w* that a particle will escape from this region in unit time; the quantity *w* is equal to the product of the oscillation frequency of the particle inside the potential well and the probability of transmission through the barrier. Since a particle located in the potential well may “leak” to the outside, the corresponding particle energy levels take on a finite width of order *hlw* (where *h* is Planck’s constant); these states are quasi-stationary.

The processes of the autoionization of an atom in a strong electric field provide an example of the tunnel effect in atomic physics. The process of the ionization of an atom in the field of a strong electromagnetic wave has attracted particularly great attention of late.

In nuclear physics the tunnel helps explain the alpha decay of radioactive nuclei. As a result of the joint action of short-range nuclear forces of attraction and electrostatic (Coulomb) forces of repulsion, an alpha particle must pass across a three-dimensional potential barrier of the type described above.

Without the tunnel effect it would be impossible for thermonuclear reactions to occur. For fusion to occur, the nuclei participating in the reaction must approach one another. They are impeded, however, by a Coulomb potential barrier, which is overcome partly as a result of the high speeds (high temperature) of such nuclei and partly as a result of the tunnel effect.

Many examples of the tunnel effect can be found in solid-state physics. Thus, field emission—that is, the emission of electrons from a metal or semiconductor under the influence of a strong electric field—occurs through tunneling. The tunnel effect accounts for various phenomena in semiconductors placed in a strong electric field; the tunnel diode is an important example in this connection. The migration of valence electrons in a crystal lattice involves tunneling (*see*SOLID). In the Josephson effect electron pairs tunnel through a thin insulating barrier between two superconducting materials.

### REFERENCES

Blokhintsev, D. I.*Osnovy kvantovoi mekhaniki*, 4th ed. Moscow, 1963.

Landau, L. D., and E. M. Lifshits.

*Kvantovaia mekhanika: Nereliati-vistskaia teoriia*, 3rd ed. (

*Teoreticheskaia fizika*, vol. 3.) Moscow, 1974.

D. A. KIRZHNITS